Geometric Representations of Interacting Maps

Abstract

Tropical geometry is a kind of dynamical scale transform which connects automata with real rational dynamics. Real rational dynamics are deeply studied from global analytic viewpoints. On the other hand, automata appear in various contexts in topology, combinatorics, and integrable systems. In this paper we study the analysis of these materials passing through tropical geometry. In particular we discover a new duality on the set of automata which arise from the projective duality in algebraic geometry.

1. Introduction

Motivated by the phenomena in molecular biology, in [1] we have formulated random interaction systems by use of families of maps on intervals, which consist of infinite families of compositions between them. By wasting detailed and extracting more rough information by use of projections, one can produce automata. This construction covers some discrete integrable systems which possess solitons. This paper is a trial to represent creation of some macroscopic patterns which arise from random and micro dynamics.

Let 𝑓∶[0,1]→[0,1] be a map, and consider its iteration {𝑓^𝑛(𝑥)}^∞_𝑛=0. This will behave very randomly and will touch sensitively with respect to the initial points 𝑥∈[0,1] (see [2]).

Let 𝑋₂={(𝑎₀,𝑎₁,…)∶𝑎_𝑖∈{0,1}} be the set of one-sided sequences, and take two maps {𝑓₀,𝑓₁}. Let us generalize the iteration of a single map to random compositions by two maps. Let us choose an element 𝑘=(𝑘₀,𝑘₁,…)∈𝑋₂. Then, correspondingly one obtains families of maps {ℎ^𝑛∶[0,1]→[0,1]}_𝑛=0,1,… on the interval by compositions ℎ^𝑛(𝑥)=𝑓_𝑘𝑛∘⋯𝑓_𝑘0(𝑥).

Using them, one can construct continuous maps Φ(𝑥)=Φ(𝑓₁,𝑓₂)(𝑥)∶𝑋₂→𝑋₂ for each point 𝑥∈[0,1], which are called the interaction maps. These are defined by use of projections on the interval, and so their values are determined by which subinterval ℎ^𝑛(𝑥) lies on for each 𝑛. Thus in order to determine a value Φ(𝑥)(𝑘), it is enough to know rough values of them, rather than rigorous ones on [0,1], even though one is required to have information on ℎ^𝑛(𝑥) for all 𝑛. So Φ(𝑥) is a map in a macroscopic scale compared with the micro interaction {ℎ^𝑛(𝑥)}. This might explain one aspect of a very simple mechanism to create patterns in macro scale from random micro dynamics. This method is immediately generalized to use families of maps on intervals {𝑓_𝑖}_𝑖, and one also obtains interaction maps Φ(𝑓₁,…,𝑓_𝑘) by the same way.

So far, we have known that many important cell automata can be expressible by interaction systems as above (see [3, 1.A]). In the first part of this paper, we study geometric properties of such cell automata which include the box and ball system (BBS), Lotka-Volterra cell automaton, and lamplighter automaton. For example, we will construct assignments from BBS flows to braid groups and extensions of BBS actions on compactified spaces passing through group actions on trees. We will see some relation between LV cell automaton and the lamplighter automaton, and generalize the latter to obtain more group actions on trees using such relation.

In the middle part, we study connections of automata with complex geometry. In real algebraic geometry, some geometric mappings from complex planes to the real ones were discovered, by taking coordinatewisely absolute values and their logarithms (see [4, 5]). This connects algebraic varieties with piecewise linear maps and is called the tropical geometry. On the other hand, in [6] several relations of pl maps with automata are studied. For example, the LV cell automaton is given by a family of pl maps. Passing through the above operations, in this paper we will associate a family of polynomials from an automaton. Using the projective duality on projective varieties and the above assignment, we will discover a duality between cell automata. In fact, for an automaton 𝐴, we will associate a dual one 𝐴^∨, and call it as the projectively dual automaton. We will calculate some examples in the case of curves. Notice that duality on the Mealy automata was already introduced by exchanging the role of the exit and the transition functions mutually (see [7, 8]).

In general, automata are represented by families of interaction maps. Thus direct assignments from families of maps to these geometric objects will show geometric representations of interaction systems.

In the last part, we construct infinite families of graphs from finite families of maps, points, and symbolic sequences, and we call them the interaction graphs. We will construct these graphs so that they represent some dynamics of interacting states in mesoscopic scale. The central dogma in molecular biology tells us that proteins are products of various interaction systems in mesoscopic scale, beginning from DNA. From geometric point of view, one would like to obtain a space and an automorphism on it from such interacting data, which will represent more macro features. In analogy to the central dogma as above, the space will be compared with protein or polymer, and orbits of the automorphism correspond to states on it, since functions of proteins are determined by the shapes of themselves.

A biological space-form problem is to construct a space 𝑉 and an automorphism 𝐴 on it so that the orbits {𝑥,𝐴(𝑥),𝐴²(𝑥),…}⊂𝑉 are induced from the dynamics of the interaction graphs. In this paper we will formulate and address a space-form problem from toric variety viewpoint, passing through several hierarchies of dynamics.

Throughout this paper, our basic direction is to study geometric properties of them as macro objects, which are induced by scaling transform of dynamics by compositions and iterations of families of maps as the micro one. In some particular cases of families of maps which include some piecewise linear maps, the interaction maps induce some automata.

On the other hand, in general, maps on intervals are very flexible objects and their dynamics are very random. One of our main aims in further development will be to study macro properties of these dynamics for families of maps which are near such special types of maps. Namely, let {𝑓_𝑖}_𝑖 be a family as above so that the corresponding interaction system induces some automaton 𝐴. Let us take any geometric object 𝐺=𝐺({𝑓_𝑖}_𝑖) arising from the interaction system. We have several examples of 𝐺 below. Since 𝐺 passes through 𝐴, its structure will hold some rigid properties.

Let us take another family of maps {𝑔_𝑖}_𝑖 which is sufficiently near the original {𝑓_𝑖}_𝑖, where the corresponding interaction system for {𝑔_𝑖}_𝑖 may not induce any automata in general. We would like to study geometric properties of 𝐺({𝑔_𝑖}_𝑖) by comparing with 𝐺({𝑓_𝑖}_𝑖). This might give one direction of study to understand mechanisms of creation of patterns. In particular, several stability properties of 𝐺 under small deformation of these maps will be particularly of interest for us.

Now let us describe the contents of the paper more concretely.

The box and ball system (BBS) is a dynamical system on the set of finite subsets in 𝐙 (see [9]). Let us describe the dynamics shortly. For {𝑖₁,…,𝑖_𝑙}⊂𝐙 with 𝑖₁<𝑖₂<⋯<𝑖_𝑙, let us imagine that a ball occupies in each position 𝑖_𝑚 for 𝑚=1,…,𝑙. These balls will be moved by the following rule; the ball 𝑖₁ in the most left-hand side is moved to some 𝑖₁<𝑗₁ which is the most left-hand side in {𝑖₁+1,𝑖₁+2,…}⧵{𝑖₂,…,𝑖_𝑙}. We repeat the same procedure for 𝑖_𝑘 with 2≤𝑘≤𝑙. When this procedure is finished for 𝑖_𝑙, then we are done.

Let Σ⁰₂ be all of the finite subsets in 𝐙. Then, the above procedure is expressed as 𝑇∶Σ⁰₂≅Σ⁰₂. (1.1) Let Σ⁰₂(𝑁)⊂Σ⁰₂ be the sets of 𝑁-subsets in 𝐙. Then, since BBS preserves the number of the balls, 𝑇 is a map as 𝑇∶Σ⁰₂(𝑁)≅Σ⁰₂(𝑁).

Following our expression of interaction, in Section 2, we will describe the BBS system by an interaction of maps for some family of continuous maps.

It is known that the BBS flows by the dynamics of 𝑇 contain solitons. Among dynamical properties of solitons, the relative positions of the individual waves and how these waves pass through the others will be in the most important structures. In Section 3 we will represent such information geometrically by using the braid groups. Elements in braid groups are certainly representing such situation. In order to eliminate infinitely many ambiguities, in this paper we will use quotient groups 𝐵_𝑛 of the braid groups. There are many of them, and in a special case it is a subgroup of the mapping class group of finite index. Each 𝜎 admits an index among subsets in {1,…,𝑁−1}, which is determined by 𝑇^𝑡(𝜎) near 𝑡=±∞. We will associate a quotient braid group with respect to individual index. Then, we obtain the canonical maps 𝐵∶Σ⁰₂(𝑁)→𝐵_𝑁 (1.2) which we call the braiding map. 𝐵(𝜎) is constructed from the dynamics of iterations {…,𝑇⁻¹,𝜎,𝑇(𝜎),𝑇²(𝜎),…}. (1.3) Scattering process of BBS is described in [10] by using the combinatorial 𝑅-matrix. Using such direction, one may obtain some invariants of the dynamics 𝑇.

Let Σ₂ be the set of two-sided sequences with the alphabets {0,1}. Then, Σ⁰₂ can be regarded as a subset Σ₂. In Section 5, we study geometric properties of the BBS map 𝑇. For example, we see that it can be extended as 𝑇∶Σ₂→Σ₂.

The Lotka-Volterra cell automaton is given by the equation (see [9]) 𝑉_𝑛^𝑡+1−𝑉^𝑡_𝑛=max(0,𝑉^𝑡_𝑛+1−𝐿)−max(0,𝑉^𝑡+1_𝑛−1−𝐿). (1.4) This is obtained from the original Lotka-Volterra equation by making discretizations twice. It is known that this possesses solitons which are induced from the ones of the difference LV solitons (see [3]). BBS is isomorphic to the Lotka-Volterra cell automaton. In fact, there is an explicit procedure to construct such isomorphism. Thus one obtains an injection Σ⁰₂↪Σ⁰_∞ which assigns flows of the BBS to the corresponding dynamics of the LV. We will see that it can be extended on some partially compactified space. In Section 4, we study structures of path spaces of the set of cell automata, which arose from the relation of LV-CA with BBS.

The lamplighter group is well known as an automata group in geometric group theory (see [11]), which admits an action on the rooted binary tree. In Section 5, we find that in a special case the LV cell automaton is in fact a transition function for the lamplighter automaton.

Lemma 1.1. Suppose that the initial sequence (𝑎₀,𝑎₁,…) consists of only {0,1} entries. Then, the degeneration of the LV cell automaton is the same as the transition function 𝜙 of an automaton whose group is isomorphic to the lamplighter group.

Thus using the LV cell automaton as transition functions, one can construct a family of automata groups which we call LV cell automata groups. They can act on the boundary 𝜕𝑇^∗_∞≡⋃_𝑖𝜕𝑇^∗_𝑖 of the rooted infinite tree.

In Section 6, we study automata from complex geometry point of view. Tropical geometry connects algebraic geometry with automata. There is a kind of scale transform Φ_𝑡 with 𝑡∈(1,∞) by taking absolute values and taking the conjugation by Log_𝑡 (see [4, 12]). To a pair of parameterized polynomials (𝑓¹_𝑡,𝑓²_𝑡), both lim_𝑡→∞Φ_𝑡(𝑓¹_𝑡) and lim_𝑡→∞Φ_𝑡(𝑓²_𝑡) become piecewise linear maps. In general, the equation Φ_∞(𝑓¹_∞)=Φ_∞(𝑓²_∞) represent complicated dynamics, but in some cases it gives automata. This is the point where we find importance to study stability of the dynamics. We study this aspect in Section 6.3.2.

On the other hand, the above process is invertible in the sense that one can associate pairs of parameterized polynomials (𝑓¹_𝑡,𝑓²_𝑡) from automata 𝐴. Now using these polynomials, one can obtain spaces as follows.

Definition 1.2. The associated affine hypersurfaces are parameterized family of hypersurfaces given by the following equations: 𝑉(𝐴)_𝑡={𝐳∈𝐂^𝑁∶𝑓¹_𝑡(𝐳)=𝑓²_𝑡(𝐳)}. (1.5)

One important reason to consider such varieties comes from projective duality on algebraic varieties. Let us denote by 𝑋^∨ the projective dual of an algebraic variety 𝑋. Suppose that the projective duals of 𝑉(𝐴)_𝑡 are also hypersurfaces. The parameterized families of polynomials Δ(𝐴)_𝑡 which define these hypersurfaces are called 𝐴-discriminant.

Again by taking the above scaling limit in tropical geometry, one obtains another automaton 𝐴^∨. Thus passing through duality in complex geometry, there is an assignment between automata: 𝐴→𝐴^∨. (1.6) We call 𝐴^∨ the projectively dual automaton.

We have calculated in the case of some curves. Let 𝑎≥2, 𝛼, and 𝑐 be integers. Then, we have the following.

Proposition 1.3. Consider the following: [max{𝑎𝑢_𝑛,𝛼+𝑎𝑢_𝑛+1}=𝑐]^∨𝑎=max{𝛼𝑎−1(𝑐−𝑎𝑎)+𝑢𝑎−1^𝑛+1,𝑎𝑐+𝑎𝑎−1𝑢𝑎−1^𝑛}=𝑐. (1.7)

In micro level, molecular interactions occur by covalent or hydrogen bonds where electrons of molecules share their orbitals. Inspired on this aspect, in Section 7 one formulates interaction systems of families of maps by constructing some graphs.

Let us take two interval maps 𝑓₀,𝑓₁∶[0,1]→[0,1] and let Φ(𝑥,𝑓₀,𝑓₁)∶𝑋₂→𝑋₂ be the interaction map. Let us choose another map 𝑑∶[0,1]→[0,1]. In the light of orbitals stated above, if the projection of the iterations {𝑑^𝑛(𝑧)}_𝑛 coincides with Φ(𝑥,𝑓,𝑔)(𝑘) in 𝑋₂, then we construct a marked oriented edge as (𝑓,𝑥)−−−−→^(𝑔,𝑘)(𝑑,𝑧). (1.8)

Let us fix families of maps {𝑓₁,…,𝑓_𝑘}, points {𝑥₁,…,𝑥_𝑙}⊂[0,1], and {𝑎(𝑖,𝑗,ℎ)}^{𝑖,𝑗=𝑘,ℎ=𝑙}_{𝑖,𝑗,ℎ=1}∈𝑋₂. By the above way, we obtain the corresponding oriented marked finite graph 𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)}). (1.9) We call them the interaction graphs.

One can interpret these graphs to represent states of the system consisted by the triple ({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)}).

Let us choose (𝑓_𝑖,𝑓_𝑗,𝑥_ℎ). Then, we have the corresponding interaction map Φ(𝑓_𝑖,𝑓_𝑗,𝑥_ℎ)∶𝑋₂→𝑋₂. In particular one can obtain other elements as follows: 𝑎(𝑖,𝑗,ℎ)₂≡Φ(𝑓_𝑖,𝑓_𝑗,𝑥_ℎ)(𝑎(𝑖,𝑗,ℎ))∈𝑋₂. (1.10) By this way one can obtain another interaction graph: 𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)₂}). (1.11)

Let us denote by 𝔊({𝑓_𝑖},{𝑥_𝑗}) the set of interaction graphs with fixed families of maps and points. The numbers of vertices are all the same in any element in this. Notice that this is a finite set. Then, by the above procedure one has obtained the following map: Φ_∗∶𝔊({𝑓_𝑖},{𝑥_𝑗})→𝔊({𝑓_𝑖},{𝑥_𝑗}). (1.12)

By iterating this procedure, one obtains an infinite family of interaction graphs 𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)}), 𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)₂}), 𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)₃}),…. One can regard that this family of interaction graphs might represent dynamics of states of the micro interaction systems, and according to our principle at the beginning of the introduction, one may induce some macro patterns from them. In this paper we will induce some hierarchies of combinatoric objects arising from such family of graphs.

Let 𝔊 be the set of finite graphs, and let 𝐹∶𝔊({𝑓_𝑖},{𝑥_𝑗})→𝔊 be the forgetful map. Passing through 𝐹, one obtains a family of finite graphs: 𝐺₁≡𝐹(𝐺₁),𝐺₂…⊂𝔊,𝐺_𝑡=𝐺({𝑓_𝑖},{𝑥_𝑗},{𝑎(𝑖,𝑗,ℎ)_𝑡}), (1.13) which we call just the associated graphs. Any 𝐺_𝑖 has the same number of vertices 𝑁. Thus there is a finite number of finite graphs {𝐻₁,…,𝐻_𝑚} so that each 𝐺_𝑖 coincides with one of {𝐻_𝑗}.

We say that a family of finite graphs is strongly regular if they have the same number of edges as others.

Let 𝐺 be a finite graph. Then, the associated configuration 𝑎∈𝐙^𝑚 in combinatorics is determined. Thus one obtains another family of configurations: 𝑎₁,𝑎₂,𝑎₃,…⊂𝐙^𝑚, (1.14) which we call the transcripted configurations.

For each configuration 𝑎∈𝐙, one obtains an ideal 𝐼_𝑎⊂𝐂[𝑦₁,…,𝑦_𝑚] which is called the toric ideal. Thus associated to the family of configurations, one obtains the corresponding family of ideals: 𝐼₁,𝐼₂,…,𝐼_𝑗⊂𝐂[𝑦₁,…,𝑦_𝑚𝑗], (1.15) which we call the associated ideals.

For each ideal 𝐼⊂𝐂[𝑦₁,…,𝑦_𝑚], one obtains a complete fan over 𝐑^𝑚 and the corresponding toric variety 𝑋_𝐼⊂𝐂𝐏^𝑚−1. The fan is called the Gröbner fan.

Thus corresponding to the associate ideals, one obtains a family of toric varieties: 𝑋₁,𝑋₂,…⊂𝐂𝐏^𝑚−1,𝑚=sup{𝑚₁,𝑚₂,…}. (1.16) We call the sequence as the translated toric variety. When the associated graphs are strongly regular, then all 𝑋_𝑗 have the same dimension.

Let us have more abstract setting. Let 𝑉 be an algebraic variety with the affine coordinates 𝑉_𝑖 defined by an ideal 𝐽_𝑖. Let us take an automorphism 𝐴 on 𝑉. We say that an affine coordinate {(𝑉_𝑖,𝐽_𝑖)}^𝑚_𝑖=1 is an (stable) algebraic Markov partition for 𝐴 if, for each 𝑖, there is some 𝑗 so that 𝐴(𝑉_𝑖)⊂𝑉_𝑗 (1.17) holds.

We say that the associated ideals {𝐼_𝑖}_𝑖 are regular if they have the same dimension as others. Let us put ⋃_𝑖𝐼_𝑖={𝐽₁,…,𝐽_𝑘}. Now we have one realization of the space-form problem.

Definition 1.4. Let ({𝑓_𝑖}_𝑖,{𝑥_𝑗}_𝑗,{𝑎(𝑖,𝑗,ℎ)}) be interaction data, and suppose that the associated ideals 𝐼=(𝐼₀,𝐼₁,…) are regular. The sequence is called a symbolic flow of an automorphism if there is an algebraic Markov partition for (𝑉,𝐴) with an affine coordinate {(𝑉_𝑖,𝐽_𝑖)}^𝑘_𝑖=1 and some 𝑥∈𝑉 so that its orbit {𝐴^𝑛(𝑥)}_𝑛=0,1,… corresponds to the sequence.

We would like to call such pair (𝑉,𝐴) a prohedron (which comes from “proteiform’’). We will see by an easy example that combinatorics of the interaction graphs will reflect existence of such pairs.

The author would like to thank the referee for giving him the useful suggestions and comments.

2. Automata and Interaction by Families of Maps

2.1. Classes of Automata

In this paper we treat some automata whose classes we will clarify below.

Let 𝐴 be a finite set called the alphabet, and let 𝑄 be a set called the state set.

Let us introduce the most general form of automata here. Let 𝑚,𝑙≥0 be integers. A bounded automaton 𝐀 over 𝐴 is given by two functions: 𝜓∶𝑄×𝐴^𝑚+𝑙+2→𝐴,𝜙∶𝑄×𝐴→𝑄. (2.1)

A flow of the sets of strings of one-sided infinite length {𝑘_𝑡=(𝑘^𝑡₀,𝑘^𝑡₁,…)∶𝑘^𝑡_𝑖∈𝐴}_𝑡≥0 (2.2) is determined by 𝐴 if {𝑘^𝑡_𝑁}_{𝑁,𝑡≥0} satisfy the following relations: 𝑘_𝑖^𝑡+1=𝜓(𝑞_𝑖^𝑡+1,𝑘^𝑡_𝑖−𝑙,…,𝑘^𝑡_𝑖+𝑚,𝑘^𝑡+1_𝑖−1),𝑞^𝑡_𝑖=𝜙(𝑞^𝑡_𝑖−1,𝑘^𝑡_𝑖). (2.3)

We regard that they grow time evolutionally as 𝑘₀→𝑘₁→⋯→𝑘_𝑡→⋯.

One can similarly consider the two-sided case {𝑘_𝑡=(…,𝑘^𝑡_−𝑛,…,𝑘^𝑡₀,𝑘^𝑡₁,…)∶𝑘^𝑡_𝑖∈𝐴}_𝑡≥0. (2.4)

2.1.1. Mealy Automaton (see [7, 8])

A Mealy automaton 𝐀 over 𝑋 is given by two functions: 𝜓∶𝑄×𝐴→𝐴,𝜙∶𝑄×𝐴→𝑄. (2.5)

(A) One-Sided Case
Let 𝑋_𝐴={𝑘=(𝑘₀,𝑘₁,…)∶𝑘_𝑖∈𝐴} (2.6) be all of the set of strings of one-sided infinite length. Then, each 𝑞∈𝑄 induces a continuous map 𝐴_𝑞∶𝑋_𝐴→𝑋_𝐴 (2.7) given by 𝐴_𝑞(𝑘₀,𝑘₁,…)=(𝑘^′₀,𝑘^′₁,…), where 𝑘^′_𝑖 are inductively defined with 𝑞₋₁=𝑞 as follows: 𝑘^′_𝑖=𝜓(𝑞_𝑖,𝑘_𝑖),𝑞_𝑖=𝜙(𝑞_𝑖−1,𝑘_𝑖). (2.8)
One can check easily that Mealy automata can be obtained by choosing 𝜓∶𝑄×𝐴→𝐴 by 𝑘_𝑖^𝑡+1=𝜓(𝑞_𝑖^𝑡+1,𝑘^𝑡_𝑖) and 𝜙 are the same.
Let 𝑋^𝑁_𝐴={𝑘^∗=(𝑘₀,𝑘₁,…,𝑘_𝑁)∶𝑘_𝑖∈𝐴} be the set of words of length 𝑁 with alphabet 𝑋. Then, 𝐴_𝑞 restricts the action as 𝐴_𝑞∶𝑋^𝑁_𝐴→𝑋^𝑁_𝐴 for all 𝑁∈{0,1,2,…}.
Let 𝑚=|𝐴| and let 𝑇_𝑚 be the rooted 𝑚-tree. The set of all vertices of 𝑇_𝑚 can be identified with 𝑋^∞_𝐴≡⋃_𝑁𝑋^𝑁_𝐴. Thus 𝐴_𝑞 gives the following action: 𝐴_𝑞∶𝑇_𝑚→𝑇_𝑚. (2.9)
Let us say that 𝐴 is invertible (see [7]) if 𝜓(𝑞,)∶𝐴≅𝐴 are one-to-one onto for all 𝑞∈𝑄. An invertible automaton 𝐴 gives automorphisms 𝐴_𝑞∶𝑇_𝑚≅𝑇_𝑚, and the group generated by the set of states is denoted by 𝐺(𝐴): 𝐺(𝐴)=gen{𝐴_𝑞∶𝑇_𝑚≅𝑇_𝑚∶𝑞∈𝑄}. (2.10)

(B) Two-Sided Case
Let 𝐴={0,1,…,𝐿−1} and let 𝑆 be a finite set. Let us consider the two maps 𝜓∶𝑆×𝐴→𝐴,𝜙∶𝑆×𝐴→𝑆 (2.11) equipped with the initial state 𝑞∈𝑆. These data give a structure of Mealy automaton 𝐀.
Let us put the set of the two-sided sequences by Σ_𝐿={(…,𝑣_−𝑛,…,𝑣₀,…)∶𝑣_𝑖∈𝐴}. Let 𝜎=(…,𝑣_−𝑛,…,𝑣₀,…) be an infinite sequence by {0,…,𝐿−1} such that 𝑣_𝑛=𝑣_−𝑛=0 holds for all sufficiently large 𝑛≫0. Let us denote the set of such sequences by Σ⁰_𝐿={(…,𝑣_−𝑛,…,𝑣₀,𝑣₁,…)∶𝑣_𝑖∈𝐴,𝑣_𝑛=0∀large|𝑛|≫0}. (2.12) Notice that Σ⁰_𝐿 are shift invariant.
Let us say that 𝐀 is semiproper if 𝜓(𝑞,0)=0 is satisfied.
If 𝐀 is semi-proper, then it induces a continuous map 𝐴_𝑞∶Σ⁰_𝐿→Σ_𝐿,𝐴_𝑞(…,𝑣_𝑛,𝑣_𝑛+1,…)=(…,𝑣^′_𝑛,𝑣^′_𝑛+1,…) (2.13) given by the same rule as that of the one-sided case. Namely, let us take (…,𝑣_𝑛,𝑣_𝑛+1,…)∈Σ⁰_𝐿, and choose sufficiently small 𝑚₀ so that 𝑣_𝑚=0 holds for all 𝑚≤𝑚₀. Then, we define inductively as 𝑣^′_𝑚0+𝑘=𝜓(𝑠_𝑘,𝑣_𝑚0+𝑘),𝑠_𝑘=𝜙(𝑠_𝑘−1,𝑣_𝑚0+𝑘), (2.14) where we put 𝑣^′_𝑚=0 for all 𝑚≤𝑚₀. This is independent of choice of 𝑚₀ and gives an assignment 𝐴_𝑞∶(…,𝑣_𝑚,𝑣_𝑚+1,…)→(…,𝑣^′_𝑚,𝑣^′_𝑚+1,…). (2.15)
Let 𝐀 be a Mealy automaton which is semi-proper. If it induces a continuous map 𝐴_𝑞∶Σ⁰_𝐿→Σ⁰_𝐿, (2.16) then we say that 𝐀 is proper.

2.1.2. Cellular Automaton

The cell automata we treat here require only the initial states. Let 𝐴=𝑄, and take a function 𝜓∶𝐴³→𝐴. Then, each 𝑞∈𝑄=𝐴 induces a continuous map 𝐴_𝑞∶𝑋_𝐴→𝑋_𝐴 (2.17) given by 𝐴_𝑞(𝑘₀,𝑘₁,…)=(𝑘^′₀,𝑘^′₁,…), where 𝑘^′_𝑖 are inductively defined with 𝑘^′₋₁=𝑞: 𝑘^′_𝑖=𝜓(𝑘_𝑖,𝑘_𝑖+1,𝑘^′_𝑖−1). (2.18) So the flows {𝑘_𝑡}_𝑡 satisfy the relations 𝑘_𝑖^𝑡+1=𝜓(𝑘^𝑡_𝑖,𝑘^𝑡_𝑖+1,𝑘^𝑡+1_𝑖−1).

2.2. Interacting Maps

Let us take the two interval maps 𝑓₀,𝑓₁∶[0,1]→[0,1] (2.19) and consider their iterations as 𝑂₀(𝑥)={𝑓^𝑘₀(𝑥)}_𝑘=0,1,…,𝑂₁(𝑥)={𝑓^𝑘₁(𝑥)}_𝑘=0,1,…. (2.20) We call them the oscillations (see [3]).

Let us define interaction of these orbits below. For this, let 𝑋₂ be the set of one-sided sequences with two alphabets {0,1} as follows: 𝑋₂={(𝑘₀,𝑘₁,…)∶𝑘_𝑖∈{0,1}}. (2.21) For each element 𝑘=(𝑘₀,𝑘₁,…)∈𝑋₂, we associate a family of maps {ℎ^𝑚}_𝑚=0,1,… with ℎ^𝑚∶[0,1]→[0,1] by the random iterations ℎ^𝑚(𝑥)≡𝑓_𝑘𝑚∘𝑓_𝑘𝑚−1∘⋯∘𝑓_𝑘0(𝑥). (2.22)

Let 𝜋∶[0,1]→{0,1} (2.23) be the measurable map given by 𝜋([0,1/2))≡0 and 𝜋((1/2,1])≡1. Then, for each 𝑥∈[0,1], one can compose {ℎ^𝑚(𝑥)}_𝑚 with 𝜋 and obtain another element for a.e. 𝑥 as follows: 𝑘^′≡𝜋((ℎ⁰(𝑥),ℎ¹(𝑥),…))≡(𝜋∘ℎ⁰(𝑥),𝜋∘ℎ¹(𝑥),…)∈𝑋₂. (2.24) This assignment gives a map from each element 𝑘 to 𝑘^′. We denote this map Φ(𝑥,𝑓₀,𝑓₁)∶𝑋₂→𝑋₂ (2.25) by Φ(𝑥,𝑓,𝑔)(𝑘)=𝑘^′≡𝜋((ℎ⁰(𝑥),ℎ¹(𝑥),…)) and call it the interaction map.

Let us introduce a generalization of the interaction. Let us fix a set of alphabets {0,1,…,𝐿} and choose a family of maps {𝑓_𝑖,𝑗}_{𝑖,𝑗=0,1,…,𝐿} and 𝑘∈𝑋_𝐿+1. Then, we define inductively ℎ^𝑚(𝑥)=𝑓_{𝑘𝑚,𝑘𝑚+1}∘ℎ^𝑚−1(𝑥),ℎ⁻¹(𝑥)≡𝑥. (2.26) We say that {ℎ^𝑚(𝑥)}_𝑚 is a two-step interaction.

Let 𝜋∶[0,1]→{0,1,…,𝐿} be the projection. Then, as before one obtains a map Φ(𝑥)∶𝑋_𝐿+1→𝑋_𝐿+1,Φ(𝑥)(𝑘)=𝑘^′,𝑘^′_𝑚=𝜋∘ℎ^𝑚(𝑥). (2.27) Let Φ∶𝑋_𝐿+1→𝑋_𝐿+1 be a map, and take an initial condition (𝑘⁰₀,𝑘⁰₁,𝑘⁰₂,…) (2.28) and a boundary condition (𝑘¹₀,𝑘²₀,𝑘³₀,…). (2.29) Then, we inductively put (𝑘^𝑡₀,𝑘^𝑡₁,…)=Φ(𝑘₀^𝑡−1,𝑘₁^𝑡−1,…)=Φ^𝑡(𝑘⁰₀,𝑘⁰₁,𝑘⁰₂,…)∈𝑋_𝐿+1. (2.30) We call (𝑘^𝑡₀,𝑘^𝑡₁,…) as the flow of Φ.

In Section 6, we consider the flow of the Lotka-Volterra cell automaton by the iteration of the interaction map for some family of interval maps. LV cell automaton is isomorphic to the BBS system described below. We will study the isomorphism in Section 4.

2.3. BBS System

Let Σ₂ be the set of two-sided sequences with two alphabets, and consider its subsets Σ⁰₂⊂Σ₂: Σ⁰₂={(…,𝑣_−𝑛,…,𝑣₀,𝑣₁,…)∈Σ₂∶𝑣_𝑛=0∀large|𝑛|≫0}. (2.31) Notice that it is shift invariant.

Let us describe the dynamical system 𝑇∶Σ⁰₂≅Σ⁰₂ called BBS below (see [9]). One can canonically identify Σ⁰₂ with the set of ordered integers Σ⁰₂≅𝑂={𝑖₁<𝑖₂<⋯<𝑖_𝑚∶𝑖_𝑘∈{…,0,1,…},𝑚∈ℕ} (2.32) by (…,𝑣_−𝑛,…,𝑣_𝑛,…)→{𝑛∶𝑣_𝑛=1} as in the introduction.

Let us choose an element 𝜎∈Σ⁰₂ and let (𝑖₁<𝑖₂<⋯<𝑖_𝑚) be the corresponding ordered integers. Let 𝑇(𝜎)₁=(…,𝑣¹_−𝑚,…,𝑣¹₀,𝑣¹₁,…)∈Σ⁰₂ (2.33) be another element defined as follows: let 𝑗₁≥𝑖₁ be the smallest index with 𝑣_𝑗1=0. Then, we put 𝑣¹_𝑙=𝑣_𝑙,𝑙≠𝑖₁,𝑗₁,0,𝑙=𝑖₁,1,𝑙=𝑗₁. (2.34)

Next we do the same thing for 𝑣¹_𝑖2=𝑣_𝑖2 in 𝑇(𝜎)₁ and find another smallest index 𝑗₂≥𝑖₂ with 𝑣¹_𝑗2=0. Then, we exchange 0 and 1 in 𝑣¹_𝑖2 and 𝑣¹_𝑗2 as above. The result is denoted by 𝑇(𝜎)₂.

We continue this process for 𝑖₃,𝑖₄,… until 𝑖_𝑚, and finally one obtains the desired 𝑇(𝜎)≡𝑇(𝜎)_𝑚∈Σ⁰₂.

Thus one has obtained a continuous bijective map 𝑇∶Σ⁰₂≅Σ⁰₂ (2.35) which is calld the box and ball system (BBS) (see [13]).

Let 𝑂_𝑁={𝑖₁<𝑖₂<⋯<𝑖_𝑁∶𝑖_𝑘∈{…,0,1,…}}⊂𝑂 be the subset consisted by the set of exactly 𝑁 indices. Let Σ⁰₂(𝑁)⊂Σ⁰₂ be the corresponding subsets. Then, the BBS system preserves them and induces bijections 𝑇∶Σ⁰₂(𝑁)≅Σ⁰₂(𝑁).

2.3.1. BBS as a Mealy Automaton

Let us describe the BBS map 𝑇 by an interaction of a family of interval maps. The basic method will take two steps. Firstly we will describe it by a proper Mealy automaton, and then we write down the automaton by an interaction of a family of maps. Here we will use some modified way in order to express it by a family of continuous maps. For the presentation, see also Section 4.1.

Firstly we will construct an automaton 𝐴. It will induce a map 𝑀∶Σ⁰₂(𝑁)≅Σ⁰₂(𝑁) which corresponds to 𝑇(𝜎)₁ in Section 2.3. Then, the BBS map 𝑇∶Σ⁰₂(𝑁)≅Σ⁰₂(𝑁) is given by 𝑇∣Σ⁰₂(𝑁)=𝑀^𝑁.

Lemma 2.1. For the state set 𝑆={0,1,2,3,4} with the initial state 0, there is a proper Mealy automaton 𝐀 which induces 𝑀∶Σ⁰₂≅Σ⁰₂.

Proof. Let us define 𝐀 by 𝜓∶𝑆×{0,1}→{0,1},𝜙∶𝑆×{0,1}→𝑆, (2.36) where 𝜓(𝑎,)=𝜖,𝑎=1,2,id,𝑎=0,3,4 (2.37) (𝜖∈𝑆₂ is the nontrivial element in the permutation group on {0,1}).
It is immediate to check the conclusion. This completes the proof.

Let us consider expressing BBS by an interaction of maps. Let us denote 𝜙={𝜙₀,𝜙₁} as in Table 1 and 𝜓={𝜓₀,𝜓₁}, where 𝜙_𝑖∶𝑆→𝑆 and 𝜓_𝑖∶𝑆→{0,1} for 𝑖=0,1. Let 𝜋∶[0,1]→{0,1} and 𝜋₅∶[0,1]→{0,1,2,3,4} be the canonical projections.

Table 1 

Values of 𝜙.

If the automaton above is described by compositions of a family of continuous maps {𝛼_𝑖}_𝑖=0,1 and {𝑓_𝑖}_𝑖=0,1 as 𝜙_𝑖=𝜋₅∘𝛼_𝑖 and 𝜓_𝑖=𝜋∘𝑓_𝑖, then we say that the automaton is given by an interaction of maps on intervals (see [6]).

It is immediate to see that the above 𝐀 cannot be expressible by a family of continuous maps as above. So we will modify as follows. Let us choose two permutations 𝜖₀,𝜖₁∶𝑆≅𝑆. Then, we say that 𝐴 is given by a modified interaction of maps with respect to (𝜖₀,𝜖₁), if there are families of continuous maps {𝛼_𝑖}_𝑖=0,1 and {𝑓_𝑖}_𝑖=0,1 so that the following hold:𝜙_𝑖=𝜖_𝑖∘𝜋₅∘𝛼_𝑖,𝜓_𝑖=𝜋∘𝑓_𝑖. (2.38)

When these maps can be chosen piecewisely linearly, then tropical geometry in Section 6 can be applied, since piecewise linear functions can have representations by the relative arithmetics of (max,+). BBS is also the following case.

Lemma 2.2. There are families of continuous maps and permutations so that they give the modified interaction of maps representing the proper Mealy automaton 𝐀 above. In particular the BBS automaton is given by 𝑀^𝑁=𝑀∘⋯∘𝑀 on Σ⁰₂(𝑁) for 𝑁=0,1,2,….

Proof. Let us choose permutations as 𝜖₀𝜖∶(0,1,2,3,4)→(0,2,4,1,3),₁∶(0,1,2,3,4)→(0,1,3,4,2). (2.39)
Then, we can choose piecewise linear functions 𝛼_𝑖 and 𝑓_𝑖 so that they satisfy the following properties: 𝛼₀𝑖∣[5,𝑖+15𝑖]⊂[5,𝑖+15𝛼],𝑖=0,1,2,₀3∣[5,451]⊂[5,25],𝛼₀4∣[52,1]⊂[5,35𝛼],₁𝑖∣[5,𝑖+15]⊂[𝑖+15,𝑖+25𝛼],𝑖=0,1,2,₁𝑖∣[5,𝑖+15]⊂[𝑖−15,𝑖5𝑓],𝑖=3,4,₀𝑗∣[5,𝑗+151]⊂[0,2],𝑗=0,3,4,𝑓₀𝑗∣[5,𝑗+151]⊂[2𝑓,1],𝑗=1,2,₁𝑗∣[5,𝑗+151]⊂[0,2],𝑗=1,2,𝑓₁𝑗∣[5,𝑗+151]⊂[0,5],𝑗=0,3,4. (2.40) This completes the proof.

3. Quotient of the Braid Groups and BBS System

3.1. Quotient of the Braid Groups

Let 𝐵_𝑛 be the braid group with 𝑛-strands. It has the following presentation: 𝐵_𝑛={𝑡₁,…,𝑡_𝑛−1∶𝑡_𝑖𝑡_𝑗=𝑡_𝑗𝑡_𝑖,|𝑖−𝑗|>1,𝑡_𝑖𝑡_𝑖+1𝑡_𝑖=𝑡_𝑖+1𝑡_𝑖𝑡_𝑖+1}. (3.1) It has a canonical element representing the half-twist, that is, Ω_𝑛≡(𝑡₁…𝑡_𝑛−1)(𝑡₁…𝑡_𝑛−2)⋯(𝑡₁𝑡₂)𝑡₁∈𝐵_𝑛. (3.2) The quotient group by Ω²_𝑛 is isomorphic to a subgroup of the mapping class group on (𝑛+1)-punctured sphere, which is consisted by groups which fix ∞ point and index 𝑛+1 (see [14]): 𝐵_𝑛/⟨Ω²_𝑛⟩⊂MPG_𝑛+1. (3.3)

Let us generalize it and have quotient braid groups.

Let 𝑖=1,…,𝑛−1 and 𝑘=0,…,𝑛−1, and denote subsets by [𝑖,𝑘]≡(𝑖,𝑖+1,…,𝑖+𝑘)⊂{1,…,𝑛−1}. Let 𝐵_[𝑖,𝑘]⊂𝐵_𝑛 be the subgroup generated by {𝑡_𝑖,…,𝑡_𝑖+𝑘}. By the same way, each 𝐵_[𝑖,𝑘] contains the following corresponding canonical elements: Ω_[𝑖,𝑘]∈𝐵_[𝑖,𝑘]. (3.4)

We will define two types of quotients of the braid groups using these canonical elements.

Definition 3.1. The mod 2 braid group 𝑀₂𝐵_𝑛 is given by the following: 𝑀₂𝐵_𝑛={𝑡₁,…,𝑡_𝑛−1∶𝑡_𝑖𝑡_𝑗=𝑡_𝑗𝑡_𝑖,|𝑖−𝑗|>1,𝑡_𝑖𝑡_𝑖+1𝑡_𝑖=𝑡_𝑖+1𝑡_𝑖𝑡_𝑖+1,𝑡²_𝑖}=𝐵_𝑛/gen⟨^𝑛−1_𝑖=1𝑡²_𝑖⟩. (3.5) For example, 𝑀₂𝐵₂=𝐙₂ and 𝑀₂𝐵₃ is an infinite group which has a presentation given by 𝐙₂∗𝐙₂/⟨𝑡₁𝑡₂𝑡²₁𝑡₂𝑡₁⟩.
Let us denote the quotient map by 𝜋∶𝐵_𝑛→𝑀₂𝐵_𝑛. (3.6)

Lemma 3.2. Let 𝜋(Ω²_[𝑖,𝑘])=1∈𝑀₂𝐵_𝑛 for any 𝑖,𝑘. Then, in particular one has 𝑀₂𝐵_𝑛=𝐵_𝑛/gen⟨_{[𝑖,𝑘]⊂{1,…,𝑛−1}}Ω²_[𝑖,𝑘]⟩. (3.7)

Proof. We check for 𝑛=4, and the general case will follow from this immediately.
In fact 𝜋(Ω²₄)=(𝑡₁𝑡₂𝑡₃𝑡₁𝑡₂𝑡₁)(𝑡₁𝑡₂𝑡₃𝑡₁𝑡₂𝑡₁)=𝑡₁𝑡₂𝑡₃𝑡₁𝑡₂𝑡²₁𝑡₂𝑡₃𝑡₁𝑡₂𝑡₁=𝑡₁𝑡₂𝑡₃𝑡₁𝑡²₂𝑡₃𝑡₁𝑡₂𝑡₁=𝑡₁𝑡₂𝑡₃𝑡₁𝑡₃𝑡₁𝑡₂𝑡₁=𝑡₁𝑡₂𝑡₃𝑡₃𝑡₁𝑡₁𝑡₂𝑡₁=1.
This completes the proof.

Let 𝐈⊂{1,…,𝑛−1}² be a set of subsets, and denote by gen⟨Ω_𝐼∶𝐼∈𝐈⟩ the group generated by elements Ω_𝐼. Then, we define another type of quotient braid groups by 𝐵_𝑛(𝐈)=𝐵_𝑛/gen⟨Ω_𝐼∶𝐼∈𝐈⟩. (3.8) Notice that we do not include elements like (𝑖,0) which correspond to one string. We will denote 𝜋_𝐈∶𝐵_𝑛→𝐵_𝑛(𝐈) for the projection.

Later we will have geometric meaning to divide by the twists in terms of the BBS system. In particular we will see in the following section that solitary behaviour in BBS system can be represented as an element in the quotient braid groups.

3.2. Solitary Flow

Any 𝜎∈Σ⁰₂ can be uniquely expressed by a finite set {𝐼₁,…,𝐼_𝑙}≡{(𝑖₁,𝑁₁),…,(𝑖_𝑙,𝑁_𝑙)}, (3.9) where 𝑁_𝑗≥1 and 𝑙≥0. They satisfy the following:(1)𝑣_𝑗=1 for all 𝑗∈⋃_{𝑘=1,…,𝑙}{𝑖_𝑘,𝑖_𝑘+1,…,𝑖_𝑘+𝑁_𝑘−1}, (2)𝑣_𝑗=0 holds for some 𝑗 in {𝑖_𝑘+𝑁_𝑘,…,𝑖_𝑘+1−1}, 𝑘=1,…,𝑙.

We call (𝑁₁,…,𝑁_𝑙) the index of 𝜎. Thus one may regard each 𝐼_𝑗 as one soliton.

Let 𝜎∈Σ⁰₂ and consider the BBS map 𝑇. The indices of 𝑇^𝑡(𝜎), (𝑆₁,…,𝑆_𝑚) are all constant for sufficiently small 𝑡≪0, and also for all sufficiently large 𝑡≫0, they are also constant (𝑇₁,…,𝑇_𝑚′), where Σ𝑆_𝑖=Σ𝑇_𝑗 holds. Here inequalities hold as follows: 𝑆₁≥𝑆₂≥⋯≥𝑆_𝑚,𝑇₁≤𝑇₂≤⋯≤𝑇_𝑚′. (3.10) We say that 𝜎 has its type ({𝑆_𝑖}^𝑚_𝑖=1,{𝑇_𝑗}^𝑚′_𝑗=1). (3.11)

If 𝜎 is a soliton, then the equality (𝑇₁,…,𝑇_𝑚′)=(𝑆_𝑚,𝑆_𝑚−1,…,𝑆₁) (3.12) holds. More precisely, let {𝜎^𝑡}_𝑡∈𝐙⊂Σ⁰₂ be a flow. We say that it is solitary if there are a set {𝑀₁,…,𝑀_𝑚}⊂𝐍, 1≤𝑀₁<𝑀₂<⋯<𝑀_𝑚, and families {𝑖^𝑡₁,…,𝑖^𝑡_𝑚}⊂𝐙 such that, for all sufficiently large 𝑡≫0,𝜎^−𝑡={(𝑖₁^−𝑡,𝑀₁),(𝑖₂^−𝑡,𝑀₂),…,(𝑖_𝑚^−𝑡,𝑀_𝑚𝜎)},_𝑡={(𝑖^𝑡₁,𝑀_𝑚),(𝑖^𝑡₂,𝑀_𝑚−1),…,(𝑖^𝑡_𝑚,𝑀₁)}, (3.13) where (1)𝑖^𝑡₁<𝑖^𝑡₂<⋯<𝑖^𝑡_𝑚 and 𝑖_𝑚^−𝑡<𝑖^−𝑡_𝑚−1<⋯<𝑖₁^−𝑡 hold, (2)|𝑖_𝑗^±𝑡−𝑖^±𝑡_𝑗+1|→∞ as 𝑡→∞ for all 𝑗=1,…,𝑚−1.

Thus for any 𝜎∈Σ⁰₂, the corresponding flow {𝑇^𝑡(𝜎)}_𝑡∈𝐙 is solitary.

3.3. Assignment of Braid Elements

Let us take 𝜎∈Σ⁰₂(𝑁), and let 𝑇 be the BBS map. Let us take a large 𝑡₀≫0 so that 𝑇^−𝑡0(𝜎) and 𝑇^𝑡0(𝜎) have indices (𝑆₁,…,𝑆_𝑚) and (𝑇₁,…,𝑇_𝑚′), respectively. Notice the equality 𝑁≡Σ^𝑚_𝑖=1𝑆_𝑖=Σ^𝑚′_𝑗=1𝑇_𝑗.

Let us put 𝜎₀≡𝑇^−𝑡0(𝜎), and we consider the step 𝑇(𝜎₀)_𝑖, 𝑖=1,…,𝑚, in the definition of the map 𝑇 in Section 2.2. For the step from 𝜎₀ to 𝑇(𝜎₀)₁, let us assign a natural element 𝑏₁∈𝐵_𝑁 (3.14) below, where each string corresponds to an element 1 in 𝜎₀. Namely, when the most left-hand side 1 moves into a position passing through another 𝑟 number of 1's, then 𝑡_𝑟−1⋯𝑡₁ is assigned. For example, if 𝜎₀=(…,0,1,1,0,…) moves as (…,0,0,1,1,…), then one generating element 𝑡₁ is assigned. Similarly, if 𝜎₀=(…,1,1,1,0,…) which moves as (…,0,0,1,1,1,…), then 𝑡₂𝑡₁ is assigned.

Next for the second step from 𝑇(𝜎₀)₁ to 𝑇(𝜎₀)₂, one assigns another element 𝑏₂∈𝐵_𝑁 by the same way. Continuing, one obtains another 𝑏₃,…,𝑏_𝑚.

By this way one has assigned an element 𝑏=𝑏(𝜎₀,𝑡₀)=𝑏_𝑚𝑏_𝑚−1⋯𝑏₁∈𝐵_𝑁 (3.15) which we call the braiding element.

3.3.1. Braiding Maps

Notice that 𝑏 depends on the choice of 𝑡₀ and so 𝜎₀ in the above. In fact there will be infinitely many elements which are different from each other with respect to the choice of 𝑡₀. The ambiguities arise from choices of the starting point 𝜎₀=𝑇^−𝑡0(𝜎) and the ending point 𝑇^𝑡0(𝜎). Let 𝜎 have the type ({𝑆_𝑖}^𝑚_𝑖=1,{𝑇_𝑗}^𝑚′_𝑗=1). They are essentially given by the twists of the canonical elements Ω_{[1,𝑆1−1]},Ω_{[𝑆1+1,𝑆2−1]},Ω_{[𝑆1+𝑆2+1,𝑆3−1]},…,Ω_{[𝑆1+⋯+𝑆𝑖−1+1,𝑆𝑖−1]}, (3.16) for the former and given for the latter by Ω_{[𝑇1+⋯+𝑇𝑗+1,𝑇𝑗+1−1]},Ω_{[𝑇1+⋯+𝑇𝑗+𝑇𝑗+1+1,𝑇𝑗+2−1]},…,Ω_{[𝑁−𝑇𝑚′,𝑇𝑚′−1]}, (3.17) where the rest indices all correspond to one string: 𝑆_𝑘=1(𝑘=𝑖+1,…,𝑚),𝑇_𝑙=1(𝑙=1,2,…,𝑗). (3.18)

Let us denote the set 𝐈(𝜎)={[1,𝑆₁−1],[𝑆₁+1,𝑆₂−1],…,[𝑆₁+⋯+𝑆_𝑖−1+1,𝑆_𝑖−1],[𝑇₁+⋯+𝑇_𝑗+1,𝑇_𝑗+1−1],[𝑇₁+⋯+𝑇_𝑗+1+1,𝑇_𝑗+2−1],…,[𝑁−𝑇_𝑚′,𝑇_𝑚′−1]}. (3.19) We say that 𝐈(𝜎) is the index of the BBS flow for 𝜎.

Let 𝐈⊂{1,…,𝑛−1} be a set of subsets. Then, we put Σ⁰₂(𝐈)≡{𝜎∈Σ⁰₂∶𝐈(𝜎)=𝐈}. (3.20)

Let 𝜋_𝐈 be the projection as before. Then, we define the following restricted braiding map with respect to 𝐈 as 𝐵(𝐈)∶Σ⁰₂(𝐈)→𝐵_𝑁(𝐈) (3.21) by assigning 𝜋_𝐈(𝑏), 𝑏=𝑏(𝜎₀,𝑡₀)). It is independent of choice of 𝑡₀ and gives a single map.

The above map depends on 𝐈. Below we will have another braiding map from Σ⁰₂ as a multivalued one. The target space is also obtained by the quotient of the braid group.

Let 𝜋∶𝐵_𝑁→𝑀₂𝐵_𝑁 be the projection. Thus 𝜋(𝑏)∈𝑀₂𝐵_𝑁 have ambiguity at most finitely many elements with respect to 𝑡₀. Now we define the mod 2 braiding map: 𝐵²∶Σ⁰₂(𝑁)→𝑀₂𝐵_𝑁 (3.22) as the images of all various values of 𝑡₀ for sufficiently large |𝑡₀|≫0, given by 𝜋(𝑏)≡𝑏∈𝑀₂𝐵_𝑁, where 𝑏 is as above. Thus 𝐵² is a finite multivalued map

We call 𝑏 the mod 2 braiding element.

3.4. Connected Braiding Maps

Let us take two elements as follows: 𝜎^𝑘=(…,0,𝑎_𝑖1,𝑘,…,𝑎_{𝑖𝑚𝑘,𝑘},0,…)∈Σ⁰₂,𝑘=1,2. (3.23) Choose sufficiently large 𝑀≫0. Then, we define the connected sum of 𝜎¹ with 𝜎² of length 𝑀 by 𝜎₁#_𝑀𝜎₂=(…,0,𝑎_𝑖1,1,…,𝑎_𝑖𝑚1,1,0,…,0,𝑎_𝑖2,2,…,𝑎_𝑖𝑚2,2,0,…)∈Σ⁰₂, (3.24) where 0 appears 𝑀 times in the middle.

The index of the connected sum and their union are given by 𝐈(𝜎¹,𝜎²;𝑀)=𝐈(𝜎¹)∪𝐈(𝜎²)∪𝐈(𝜎₁#_𝑀𝜎₂)⊂{1,…,𝑚₁+𝑚₂−1},𝐈(𝜎¹,𝜎²)≡_𝑀≫0𝐈(𝜎₁#_𝑀𝜎₂). (3.25) In fact 𝐈(𝜎¹,𝜎²;𝑀) is completely determined by the triple (𝐈(𝜎¹),𝐈(𝜎²),𝑀).

Let 𝐈_𝑁 be all of the set of subsets in {1,…,𝑁−1} such that each element can be an index for some 𝜎∈Σ⁰₂. Then, for large 𝑀≫0, there are maps 𝐻_𝑀∶𝐈_𝑁×𝐈_𝑁′→𝐈_𝑁+𝑁′ (3.26) which give the indices of connected sums.

Thus these induce a family of maps 𝐻(𝐈,𝐈^′;𝑀)∶𝐵_𝑁(𝐈)×𝐵_𝑁′(𝐈^′)→𝐵_𝑁+𝑁′(𝐻_𝑀(𝐈,𝐈^′)) (3.27) satisfying 𝐻(𝐈,𝐈^′;𝑀)(𝐵(𝐈)(𝜎¹),𝐵(𝐈^′)(𝜎²))=𝐵(𝐻_𝑀(𝐈,𝐈^′))(𝜎¹#_𝑀𝜎²) (3.28) for 𝐈∈𝐈_𝑁 and 𝐈^′∈𝐈_𝑁′. We will say that 𝐻(𝐈,𝐈^′;𝑀) is a connected braiding map.

Proposition 3.3. 𝐻(𝐈,𝐈^′;𝑀) is eventually period with respect to 𝑀.

This follows since the target space is divided by twists in braid groups.

4. Transformations on Cell Automata

4.1. From BBS to LV

The Lotka-Volterra cell automaton is given by the equation 𝑉_𝑛^𝑡+1−𝑉^𝑡_𝑛=max(0,𝑉^𝑡_𝑛+1−𝐿)−max(0,𝑉^𝑡+1_𝑛−1−𝐿). (4.1) This is obtained from the original Lotka-Volterra equation by taking differentiations twice. It is known that this possesses solitons which are induced from the ones of the difference LV solitons.

There is a procedure to transform BBS equation to LV cell automaton and vice versa (see [9, 13, 15]):𝐵_𝑛^𝑡+1=min{1−𝐵^𝑡_𝑛,Σ^𝑛−1_𝑖=−∞(𝐵^𝑡_𝑖−𝐵_𝑖^𝑡+1)}⇔𝑉_𝑛^𝑡+1−𝑉^𝑡_𝑛=max{𝐿,𝑉^𝑡_𝑛+1}−max{𝐿,𝑉^𝑡+1_𝑛−1}, (4.2) by changes of variables.

Let 𝑂₀=LVcellautomaton→𝑂₁→⋯→𝑂_𝑘=BBS (4.3) be a procedure of transformations. We say that it is invertible, if the procedure has its inverse 𝑂_𝑘→𝑂_𝑘−1→⋯→𝑂₀. We say that 𝑂₀ and 𝑂_𝑘 can be connected by an invertible procedure.

Lemma 4.1. LV cell automaton and BBS can be connected by an invertible procedure as follows: 𝑂₀={𝑉^𝑡_𝑛}→𝑂₁={𝑈^𝑡_𝑛}→𝑂₂={𝑆^𝑡_𝑛}→𝑂₃={𝐵^𝑡_𝑛}. (4.4)

Proof. They can be connected by three steps as follows: 𝑆^𝑡+1_𝑛+1−𝑆^𝑡_𝑛=min{0,1−𝑆^𝑡_𝑛+1+𝑆_𝑛^𝑡+1},𝑆^𝑡_𝑛=Σ^𝑛_𝑖=−∞𝐵^𝑡_𝑖,𝑈^𝑡+1_𝑛+1−𝑈^𝑡_𝑛=max{0,𝑈_𝑛^𝑡+1−1}−max{0,𝑈^𝑡_𝑛+1−1},𝑈^𝑡_𝑛=𝑆^𝑡_𝑛+1−𝑆_𝑛^𝑡+1,𝑉_𝑛^𝑡+1−𝑉^𝑡_𝑛=max{1,𝑉^𝑡_𝑛+1}−max{1,𝑉^𝑡+1_𝑛−1},𝑉^𝑛_𝑡−𝑛=𝑈^𝑡_𝑛. (4.5)
For the first transformation, one has the relations 𝐵^𝑡_𝑛=𝑆^𝑡_𝑛−𝑆^𝑡_𝑛−1. For the second, notice that 𝑆^𝑡+𝑎+1_𝑛−𝑎=0 for all sufficiently large 𝑎. Then, we have the relations Σ^∞_𝑥=0𝑈^𝑡+𝑥_𝑛−𝑥=𝑆^𝑡_𝑛+1−lim_𝑎→∞𝑆^𝑡+𝑎+1_𝑛−𝑎=𝑆^𝑡_𝑛+1. (4.6) This completes the proof.

Remark 4.2. It has been shown in [3] that LV cell automaton is given by a family of PL maps and projections on the interval [0,1]. We have seen that BBS is given by a family of PL maps, projections, and permutations in Section 2.3.

4.1.1. Deformation by Commutators

Let us consider the second step in the proof of Lemma 4.1. We express the linear transformation 𝑈^𝑡_𝑛=𝑆^𝑡_𝑛+1−𝑆_𝑛^𝑡+1 as 𝑈=𝛼(𝑆), (4.7) where we mean that 𝛼(𝑆)^𝑡_𝑛≡𝑈^𝑡_𝑛. By this way, let us express others by 𝑓(𝑆),𝑓(𝑆)^𝑡_𝑛=𝑆^𝑡_𝑛−𝑆^𝑡−1_𝑛−1,𝛽(𝑆),𝛽(𝑆)^𝑡_𝑛=𝑆^𝑡_𝑛+1−𝑆_𝑛^𝑡+1,𝛾(𝑆),𝛾(𝑆)^𝑡_𝑛=min{0,1−𝑆^𝑡−1_𝑛−1}. (4.8)

Now the second step is expressed as 𝑈=𝛼(𝑆),𝑓(𝑈)=𝛽∘𝑓(𝑆), (4.9)(∗) and the defining equation becomes 𝑓(𝑆)=𝛾∘𝛼(𝑆). (∗∗) Thus combining with (*) and (**), after the transformation, the equation changes as below: 𝑓(𝑈)=𝛽∘𝛾(𝑈). (∗∗∗)

Let us consider deforming the transformations. Let ℎ be a transformation. Then, we say that it commutes with 𝑓 if 𝑓∘ℎ=ℎ∘𝑓 (4.10) holds.

Example 4.3. Let 𝑈={𝑈^𝑡_𝑛}_𝑡,𝑛 satisfy lim_𝑖→∞𝑈^𝑡−𝑖_𝑛−𝑖=0 for each 𝑡 and 𝑛.
Let ℎ be given as ℎ(𝑈)^𝑛_𝑡=Σ⁰_𝑖=−∞𝑈^𝑡−𝑖_𝑛−𝑖, (4.11) and let 𝑓 be 𝑓(𝑈)^𝑡_𝑛=𝑈^𝑡_𝑛−𝑈^𝑡−1_𝑛−1 as above. Then, ℎ commutes with 𝑓.

Let us consider transformations given by (*) and (**) above. Then, a deformation of the transformation by a commutator ℎ is another one given below:𝑊=ℎ∘𝛼(𝑆),𝑓(𝑊)=ℎ∘𝛽∘𝑓(𝑆), (4.12) which follows from (*). When one considers the equation of the form 𝑓(𝑆)=𝛾∘ℎ∘𝛼(𝑆), (4.13) then it is changed to 𝑓(𝑊)=ℎ∘𝛽∘𝛾(𝑊).

We denote all of the set of commutators with respect to 𝑓 by 𝐶(𝑓)={ℎ∶[ℎ,𝑓]=0}. (4.14)

4.2. Spaces of Cell Automata

So far, we have considered certain types of automato in Section 2.1, mainly those whose defining equations are given by max-plus equations. These are not closed under change of variables, and so here we will generalize the classes of functions we treat.

Let us denote the set of integer-valued maps PL_𝑛={𝑓∶𝐙^3𝑛+1→𝐙,𝑓(𝑥_−𝑛,…,𝑥₋₁,𝑦_−𝑛,…,𝑦₀,𝑦₁,…,𝑦_𝑛)∈𝐙}. (4.15) Notice that there are canonical extensions of any 𝑓∈PL_𝑛 to piecewisely linear maps 𝑓∶𝐑^3𝑛+1→𝐑 just by connecting images of 𝑓 piecewisely linearly. Sometimes we will identify both of the maps. Such viewpoints will become important in Section 6.

A general automaton is given by an element of the direct limit 𝑓∈PL_∞=lim_𝑛→∞PL_𝑛, (4.16) where the defining equation of the automaton is given by 𝑇^𝑡_𝑘=𝑓({…,𝑇^𝑡_𝑘−𝑛,𝑇^𝑡_{𝑘−𝑛+1},…,𝑇^𝑡_𝑘−1},{…,𝑇^𝑡−1_𝑘−𝑛,…,𝑇_𝑘^𝑡−1,…}). (4.17)

If 𝑓 is of the form 𝑓(𝑥_−𝑁,…,𝑥₋₁,…,𝑦_−𝑛,…,𝑦₁,…,𝑦_𝑛,…), for some 𝑁, then we say that it is of (𝑁,∞) type.

If 𝑓 is of the form 𝑓(…,𝑥_−𝑛,…,𝑥₋₁,𝑦_−𝑀,…,𝑦₁,…,𝑦_𝑛,…), for some 𝑀, then we say that it is of (∞,𝑀) type.

For simplicity, we will say that they are of finite-∞ or ∞-finite types, respectively. If both cases hold, then we say that it is of (𝑁,𝑀) or of finite-finite type.

Notice that the Lotka-Volterra cell automaton is of finite-finite type and the BBS is of infinite-infinite type.

Let us denote all of the set of general automata and its subsets consisted by those of ∗-∗^′ types by 𝐆𝐀𝐮𝐭_∗-∗′⊂𝐆𝐀𝐮𝐭, (4.18) where ∗,∗^′= finite or ∞. Notice that 𝐆𝐀𝐮𝐭_∞-∞=𝐆𝐀𝐮𝐭.

We will also denote the set of general automata presented by the max-plus equations and its subsets by 𝐌𝐏_∗-∗′⊂𝐌𝐏. (4.19)

One has the following inclusions: BBS∈𝐌𝐏⊂𝐆𝐀𝐮𝐭⊃𝐆𝐀𝐮𝐭_f-f⊃𝐌𝐏_f-f∋L-VCA. (4.20)

4.3. Loop Groupoid

Let us choose two general automata 𝑂={𝑆^𝑡_𝑛} and 𝑂^′={𝑇^𝑡_𝑛} in GAut. A transformation from 𝑂 to 𝑂^′ is the one given by a change of variables: 𝑇^𝑡_𝑘=𝐹({…,𝑆^𝑡_𝑘−𝑛,𝑆^𝑡_{𝑘−𝑛+1},…,𝑆^𝑡_𝑘−1},{…,𝑆^𝑡−1_𝑘−𝑛,…,𝑆_𝑘^𝑡−1,…}) (4.21) for some linear function 𝐹. We denote the transformation by 𝑂→𝑂^′. (4.22) They are mutually invertible if both 𝑂→𝑂^′ and 𝑂^′→𝑂 hold.

Let us consider two invertible paths 𝑂₀=𝑂→𝑂₁→⋯→𝑂_𝑚=𝑂^′ and 𝑂^′₀=𝑂^′→𝑂^′₁→⋯→𝑂^′_𝑛=𝑂 in GAut. Then, the composition 𝑂=𝑂₀→⋯→𝑂_𝑚→𝑂^′₁→𝑂^′₂⋯→𝑂^′_𝑛=𝑂 (4.23) gives a loop with the origin 𝑂. Clearly two loops with the origin 𝑂 admit a natural composition, and by this operation, the set of loops Ω_𝑂={𝑂≡𝑂=𝑂₀→𝑂₁⋯→𝑂_𝑚=𝑂∶invertiblepaths} (4.24) admits a group structure.

Definition 4.4. The loop groupoid Ω is given by Ω(𝑂,𝑂^′)={𝑂=𝑂₀→𝑂₁→⋯→𝑂_𝑛∶invertiblepaths}. (4.25) Therefore, Ω(𝑂)≡Ω(𝑂,𝑂) is called the loop group.

Let 𝑂₀ be the LV cell automaton, and denote all of the set of the solutions of the equation 𝑆_LV={{𝑣^𝑡_𝑛}_𝑡,𝑛∶solutionsoftheLVcellautomaton}. (4.26) Let us take an element 𝑂∈Ω_𝑂0. Then correspondingly, there is a bijective map between solutions of the equations: 𝑂_∗∶𝑆_LV≅𝑆_LV. (4.27) We call it as an induced map associated with 𝑂∈Ω_𝑂0.

In Section 6 we construct and study an involution on some classes of GAut by use of tropical geometry and projective duality in affine algebraic varieties.

Example 4.5. Let us use the notations in Section 4.1. Let {𝐵^𝑡_𝑛} and {𝑉^𝑡_𝑛} be the BBS and LV cell automaton, respectively. In Section 4.1, one has obtained a path 𝑂={𝑂₀={𝑉^𝑡_𝑛}→𝑂₁={𝑈^𝑡_𝑛}→𝑂₂={𝑆^𝑡_𝑛}→𝑂₃={𝐵^𝑡_𝑛}}.
Let us put a general automaton 𝑇^𝑡_𝑛=𝐵^𝑡_𝑛+1−𝐵_𝑛^𝑡+1. (4.28) Then, 𝑂^′₁≡{𝑇^𝑡_𝑛}∈𝐆𝐀𝐮𝐭 is linearly invertible with {𝑈^𝑡_𝑛} by the relations 𝑈^𝑡_𝑛=Σ^𝑛_𝑖=−∞𝑇^𝑡_𝑖⇔𝑇^𝑡_𝑛=𝑈^𝑡_𝑛−𝑈^𝑡_𝑛−1. (4.29) Thus one has obtained another path 𝑂^′={𝑂₀→𝑂^′₁→𝑂₂→𝑂₃}.

5. Actions on the Boundary of Trees

5.1. Actions on the Boundary and Compactification

For any graph with marking on each edge, we denote by 𝑚(𝑒) the marking at the edge 𝑒. A tree 𝑇 is said to be biinfinite if, for any vertex 𝑣∈𝑇, it contains a geodesic real line 𝑣∈𝐑⊂𝑇.

Let 𝑇₂ be the marked biinfinite binary tree with marking {0,1} so that it contains a base path 𝑙₀∶𝐑→𝑇₂ with 𝑚(𝑙₀(𝑡))=0, 𝑡∈𝐙. Let 𝐏={𝐑→𝑇₂} be all of the set of geodesics. Then, there is a canonical inclusion 𝐏⊂Σ₂, (5.1) where Σ₂ is the set of two-sided sequences with the alphabets {0,1}. Thus Σ₂ can be regarded as a compactification of the set of all geodesics in 𝑇₂.

Let us put 𝐏⁰={𝑙∈𝐏∶𝑚(𝑙(𝑡))=0∀sufficientlylarge|𝑡|≫0}. (5.2) Then, we have the following proper inclusions: 𝐏⁰⊂𝐏⊂Σ₂. (5.3) It is easy to see that 𝐏⁰⊂Σ₂ is dense. In fact Σ₂⧵𝐏⁰ consist of elements 𝑙=(…,𝑎₋₁,𝑎₀,𝑎₁,…) such that the sets {𝑖∶𝑚(𝑎_𝑖)=1} are unbounded at least for one direction.

Now we have a natural identification (Section 2.2) Σ⁰₂≅𝐏⁰⊂Σ₂. (5.4) Then, the BBS system is described by an isomorphism Φ∶𝐏⁰≅𝐏⁰. (5.5) Notice that the only fixed point is (…,0,0,…).

Proposition 5.1. Φ can be naturally extended to the continuous map Φ∶Σ₂→Σ₂, (5.6) where Φ((…,1,1,…))=(…,0,0,…). (5.7)

Thus the extended map is not an isomorphism. More generally one has noninjective points Φ((…,0,1,0,…))=(…,0,0,1,…)=Φ((…,0,1,1,0,1,…)). (5.8) The extension also has one fixed point (…,0,…).

In Section 4.1, we have assigned the following isomorphic procedures: 𝑂₀=BBS→𝑂₁→𝑂₂→𝑂₃=LVcellautomaton, (5.9) where we denote each step by 𝐹_𝑖∶𝑂_𝑖→𝑂_𝑖+1, 𝑖=0,1,2. Each 𝑂_𝑖 admits an 𝐍 action by the flow. By the construction, we have that 𝐹₀ and 𝐹₁ both are equivariant with respect to the flow, but 𝐹₂ is not.

Let us put Σ⁰_∞≡⋃^∞_𝑚=1Σ⁰_𝑚, and Σ_∞ is defined similarly. Then, by the above procedure 𝐹∶𝑂₀→𝑂₃, we have an injection 𝐹∶𝐏⁰↪Σ⁰_∞ (5.10) which assigns the solution of the BBS to the one of LV.

Notice that 𝐹 extends as 𝐹∶𝐏→Σ_∞ with respect to the compactification of 𝐏⁰ defined above; on the other hand, it cannot be extended as 𝐹∶Σ₂→Σ_∞.

Lemma 5.2. 𝐹∶𝐏→Σ_∞ is not a surjection to the set of all solutions of the LV cell automaton.

Proof. Recall that, in the transformations above, 𝐹₀ and 𝐹₁∶𝑂₁→𝑂₂ are equivariant, and 𝐹₂ is just change of indices. Then, the result follows since there are at least two fixed points for 𝑂₂ as {(…,0,0,…),(…,1,1,…)}. (5.11)

Question 1. Let us choose a large 𝑁≫0. Describe compactification of 𝐹(𝐏⁰)∩Σ_𝑁⊂Σ⁰_𝑁.

5.2. Embedding of the Lamplighter Group

Let 𝑆 and 𝐴 be finite sets. An automata group 𝐺 is an infinite group acting on the rooted tree 𝑇^∗_𝑚, 𝑚=#𝐴, which is determined by a transition function 𝜙∶𝑆×𝐴→𝑆 and an exit function 𝜓∶𝑆×𝐴→𝐴 such that 𝜓(𝑎)∈𝑆_𝑚 is an element of the permutation group for each 𝑎∈𝐴.

Let 𝑇₂ be as in Section 5.1. Recall that the lamplighter group is an automata group which acts on the rooted binary tree 𝑇^∗₂, and it has a presentation (see [11]) 𝐺=⟨𝑎,𝛾∶𝛾²=1,[𝛾^𝑎𝑖,𝛾^𝑎𝑗]=1,𝑖,𝑗∈𝐙⟩. (5.12) The action of 𝐺 preserves each level set of 𝑇^∗₂, since it is constructed using an automaton. The automaton is given in Table 2:𝜓(𝑎)=𝜖,𝜓(𝑏)=id, (5.13) where 𝜖∈𝑆₂ is the nontrivial permutation on {0,1}. We call it the lamplighter automaton.

Table 2 

Values of 𝜙.

Let us construct another group 𝐺 which contains 𝐺 and acts on 𝑇₂. Recall that 𝑇₂ contains a line (…,0,0,…)⊂𝑇₂, and choose a base vertex ∗ on the line. By identifying ∗ with the root in 𝑇^∗₂, one can embed 𝑇^∗₂ into 𝑇₂: 𝑇^∗₂↪𝑇₂. (5.14) Now passing through this embedding, one can make 𝐺 act on 𝑇₂ by letting the same action as the lamplighter group on 𝑇^∗₂↪𝑇₂ and by putting the identity on 𝑇₂⧵𝑇^∗₂.

Now Aut 𝑇₂⊃𝐺⊃𝐺 is generated by 𝐺 and another element 𝜏. Let us describe 𝜏. Notice that each edge of 𝑇₂ is assigned by one of {0,1}. We say that an automorphism 𝑔 on 𝑇₂ preserves the marking if 𝑔(𝑒) has the same marking as the one of 𝑒 in {0,1}. Now 𝜏 is an automorphism preserving the marking and uniquely defined by the property that it shifts 𝑖-zero in ((…,0,0,…)) to 𝑖+1-one. Thus 𝜏 preserves the line. Then, 𝐺 is generated by 𝑎, 𝛾, and 𝜏.

The action of 𝐺 on 𝑇₂ has no fixed point. Moreover it is finitely generated and has quotient isomorphic to 𝐙. Thus the Bass-Serre theory suggests the following.

Question 2. Is 𝐺 an amalgam? If so, write down explicitly 𝐺₁, 𝐺₂, and 𝐴 with an isomorphism 𝐺≅𝐺₁∗_𝐴𝐺₂. (5.15)

5.3. Actions on Tree by Cellular Automata

The action of the automata group on 𝑇^∗₂ is determined by the levelwise way. In fact it is an action on each vertex of 𝑇^∗₂. This is not the case for general cellular automata, like LV cell automaton. In fact the action is determined for each path in 𝑇^∗₂, rather than points in 𝑇^∗₂. More precisely, for many automata including LV case, the image of a vertex 𝑣 by an element 𝑔∈𝐺 is determined by a neighbourhood of 𝑣.

5.3.1. LV Cell Automaton as a Transition Function

Let us consider a cell automaton Φ∶𝐍×𝐍²→𝐍, (5.16) where each 𝑈^𝑡_𝑛 is determined inductively by 𝑈_𝑛^𝑡+1=Φ(𝑈^𝑡+1_𝑛−1,𝑈^𝑡_𝑛,𝑈^𝑡_𝑛+1).

We say that {𝑈^𝑡_𝑛} degenerates with respect to Φ if there is another function 𝐺 so that it satisfies the following relation: 𝑈_𝑛^𝑡+1=Φ(𝐺(𝑈^𝑡_𝑛,𝑈^𝑡_𝑛+1),𝑈^𝑡_𝑛,𝑈^𝑡_𝑛+1). (5.17)

Recall that the lamplighter automaton has two states {𝑎,𝑏} and the alphabets {0,1}. In order to compare the LV cell automaton with the transition function of it, we regard the flow {𝑈⁰_𝑡}_𝑡→{𝑈¹_𝑡}_𝑡 as an output of a transition function. By this way let us regard that the initial flow (𝑈⁰₀,𝑈⁰₁,…), 𝑈⁰_𝑖∈{0,1} is a sequence of the alphabets and (𝑈¹₀,𝑈¹₁,…) is another sequence of the states, where one identifies 𝑎 and 𝑏 with 0 and 1, respectively.

Lemma 5.3. Suppose an initial sequence (𝑎₀,𝑎₁,…) consists of only {0,1} entries.
Then, the degeneration of the LV cell automaton by 𝐺(𝑥,𝑦)=𝑥 is the same as the transition function 𝜙 of the lamplighter automaton.

When one expresses the flow of Φ by an automaton, one puts an exit function by: 𝜓(𝑎,𝑖)=0,𝜓(𝑏,𝑖)=1,𝑖=0,1. (5.18) In particular, the corresponding continuous map on the rooted tree is not an automorphism (it is not one to one).

In general flows of the LV cell automaton take integer elements 𝑈^𝑡_𝑛∈𝐍. In order to treat these cases, let us put a sequence of states (𝑎₀,𝑎₁,…), (5.19) where 𝑎₀=𝑎 and 𝑎₁=𝑏. Let 𝑆_∞ be the group of compactly supported permutation on {0,1,…}. Then, consider an exit function 𝜓∶𝐍→𝑆_∞ (5.20) with 𝜓(0)=𝜖∈𝑆₂⊂𝑆_∞ and 𝜓(1)= id.

In general let Φ∶𝐍×𝐍²→𝐍 be a transition function, and choose an initial sequence (𝑖₀,𝑖₁,…)⊂𝐍 and any 𝑈¹₀∈𝐍.

Now we define the generalized automaton (Φ,𝜓) as follows. As an output, we will exit another sequence (𝑖¹₀,𝑖¹₁,…)⊂𝐍 as follows.

Firstly determine the sequence of the states (𝑈¹₀,𝑈¹₁,…) inductively by Φ: 𝑈¹_𝑛=Φ(𝑈¹_𝑛−1,𝑖^𝑡_𝑛,𝑖^𝑡_𝑛+1). (5.21) Then, inductively we obtain the exits by 𝑖¹_𝑛=𝜓(𝑈¹_𝑛)(𝑖⁰_𝑛). (5.22) By this way one obtains an assignment which is in fact an isomorphism: 𝑔_𝑈10∶𝑋_∞≅𝑋_∞, (5.23) where 𝑋_𝑁 are the sets of one-sided sequences with 𝑁 alphabets and 𝑋_∞ is their union. The group 𝐺(Φ,𝜓) generated by 𝑔_𝑚, 𝑚=0,1,…, is also called the generalized automata group given by (Φ,𝜓).

In the case when Φ₀ is the LV cell automaton, the equation of the transition function becomes 𝑈¹_𝑡+1=𝑖^𝑡_𝑛+max(𝐿,𝑖^𝑡_𝑛+1)−max(𝐿,𝑈^𝑡+1_𝑛−1). (5.24)

Definition 5.4. The LV cell automata group is a group acting on the boundary 𝜕𝑇^∗_∞≡⋃_𝑖𝜕𝑇^∗_𝑖 of the rooted infinite tree defined by the transition function Φ₀ and an exit function 𝜓 as above.

A generalized automata group is of bounded type by 𝑁 if it induces an action between 𝑋_𝑁=𝜕𝑇^∗_𝑁.

5.4. Quasi Actions on Trees

Let 𝐺 be a group acting on the boundary of the rooted tree 𝑇^∗₂.

We say that an element 𝛾∈𝐺 is a 𝑘₀-quasi action on 𝑇^∗₂ if there is some 𝑘₀≥0 such that, if we write 𝛾(𝑎₀,𝑎₁,…)=(𝑎^′₀,𝑎^′₁,…), then 𝑎^′_𝑀 is determined by the data (𝑎₀,𝑎₁,…,𝑎_𝑀+𝑘0) for each 𝑀=0,1,…. We denote 𝛾(𝑎₀,𝑎₁,…)_𝑀=(𝑎^′₀,…,𝑎^′_𝑀)∈𝐹₂, (5.25) where 𝐹₂ is the free group generated by {𝛼,𝛽}. We denote the word length by |𝑔|.

Suppose that 𝛾 is a 1-quasi action. Then, we have a map 𝐹_𝛾∶𝐹₂→𝐹₂,𝐹_𝛾(𝑔)=(𝛾(𝑔𝛼)_|𝑔|)⁻¹𝛾(𝑔𝛽)_|𝑔|∈𝐹₂, (5.26) which we call the differential of 𝛾. Notice that, when 𝛾 is an automorphism on the tree, then 𝐹 is the identity map.

We say that a quasi action by 𝛾 is (𝑘₀,𝑙₀)-semi Markov if there is some 𝑙₀≥0 such that 𝑎^′_𝑀 is determined by (𝑎_𝑀−𝑙0,…,𝑎_𝑀+𝑘0) for each 𝑀=0,1,….

Let 𝛾 has a 1-quasi action on 𝑇^∗₂. We say that it is bounded if there is a bounded function 𝐵∶𝐹₂→𝐹₂ so that the differential satisfies the following equality: 𝐹_𝛾(𝑔)=𝐵(𝑔𝛼)⁻¹𝐵(𝑔𝛽). (5.27)

Lemma 5.5. A bounded quasi action 𝛾 is semi-Markov.

Proof. In fact one has the equality 𝛾(𝑔𝛼)_|𝑔|𝐵(𝑔𝛼)⁻¹=𝛾(𝑔𝛽)_|𝑔|𝐵(𝑔𝛽)⁻¹. (5.28) This implies the equality 𝛾(𝑔𝛼)_{|𝑔|−𝑁}=𝛾(𝑔𝛽)_{|𝑔|−𝑁}, (5.29) where 𝑁=sup{|𝐵(𝑔)|∶𝑔∈𝐹₂}. This completes the proof.

6. Associated Algebraic Varieties

6.1. Tropical Algebra

Maslov introduced a kind of scale transform called the dequantization of the real line 𝐑. It is given by deformations of the arithmetics over the real number 𝐑, which are parameterized by a family of semirings 𝑅_𝑡 for 𝑡>1.

The multiplications and the additions are, respectively, given by 𝑥_𝑡𝑦=log_𝑡(𝑡^𝑥+𝑡^𝑦),𝑥_𝑡𝑦=𝑥+𝑦. (6.1) Notice the following particular property: 𝑥_∞𝑦≡lim_𝑡→∞𝑥_𝑡𝑦=max{𝑥,𝑦}. (6.2) This is called the tropical semiring.

Corresponding to polynomials over the usual real numbers, one obtains 𝑅_𝑡-polynomials as 𝜑_𝑡(𝑥)=(𝛼₁+𝑗¹𝑥)_𝑡⋯_𝑡(𝛼_𝑘+𝑗^𝑘𝑥)(𝑥∈𝐑^𝑛,𝑗^𝑙∈𝐙^𝑛) (6.3) whose limit 𝑡→∞ satisfies the following max-plus equation: 𝜑_∞(𝑥)=max(𝛼₁+𝑗¹𝑥,…,𝛼_𝑘+𝑗^𝑘𝑥). (6.4)

Let Log_𝑡∶(𝐂^∗)^𝑛→𝐑^𝑛 be defined as Log_𝑡(𝑧₁,…,𝑧_𝑛)≡(log|𝑧₁|,…,log|𝑧_𝑛|). (6.5)

The following can be checked easily by direct calculations, but the conclusions are highly nontrivial and interesting.

Proposition 6.1 (see [4, 12]). 𝑓_𝑡≡(log_𝑡)⁻¹∘𝜑_𝑡∘Log_𝑡∶𝐑^𝑛₊→𝐑₊ is in fact a polynomial map 𝑓_𝑡(𝑧)=Σ_𝑗𝑡^𝛼𝑗𝑧^𝑗.

Thus 𝑅_𝑡-polynomials are conjugate to the standard real polynomials by log_𝑡.

Conversely let 𝜑∶𝐑^𝑛₊→𝐑₊ be a piecewisely linear map equipped with the presentation by a max-plus equation. Then, one can associate a parameterized 𝑅_𝑡 and 𝐑 polynomials, respectively, since their presentations are determined by the coefficients 𝛼₁,…,𝛼_𝑘,𝑗¹,…,𝑗^𝑘. Later on, we will denote them, respectively, by 𝜑_𝑡 and 𝑓_𝑡 and call the associated 𝑅_𝑡 and 𝐑 polynomials with respect to 𝜑.

6.2. Tropical Maps in LV Cell Automaton

It is immediate to generalize Section 6.1 to the relative case [16].

Let 𝐹_𝑡∶𝐂^𝑁→𝐂 be a family of rational functions given by 𝐹(𝑧₁,…,𝑧_𝑁Σ)=^𝑙𝑡^𝑎𝑙𝑧^𝑗𝑙Σ_𝑚𝑡^𝑏𝑚𝑧^𝑗𝑚, (6.6) where 𝑗=(𝑗₁,…,𝑗_𝑁)∈𝐙^𝑁 and 𝑧=(𝑧₁,…,𝑧_𝑁)∈𝐂^𝑁. Then, we define the corresponding tropical polynomial 𝑓=𝑓_tr as a piecewise linear map on 𝐑^𝑁 by 𝑓_tr(𝑥₁,…,𝑥_𝑁)=max_𝑙{𝑎_𝑙+𝑗^𝑙𝑥}−max_𝑚{𝑏_𝑚+𝑗^𝑚𝑥}. (6.7) where 𝑥=(𝑥₁,…,𝑥_𝑁)∈𝐑^𝑁. Such functions are called the relative (max,+) functions.

Let us recall the Lotka-Volterra cell automaton in Section 4. We describe it as flows of the generalized interactions in Section 2.2.

Proposition 6.2 (see [3]). There is a family of smooth maps {𝑓_𝑖,𝑗}_{𝑖,𝑗=0,…,𝐿} so that the corresponding flow Φ(𝑥)^𝑡, 𝑡=0,1,…, determined by {𝑓_𝑖,𝑗}_𝑖,𝑗 gives the solutions of the Lotka-Volterra cell automaton.

Proof. Let us construct the map Φ∶{0,…,𝐿}×{0,…,𝐿}²→{0,…,𝐿},Φ(𝑘^𝑡_𝑛−1,𝑘_𝑛^𝑡−1,𝑘^𝑡−1_𝑛+1)=𝑘^𝑡_𝑛∈{0,,…,𝐿}. (6.8)
Let us put a piecewise linear map 𝑓_LV by 𝑓_LV(𝑥₁,𝑥₂,𝑥₃)=𝑥₁+max{𝐿₀,𝑥₂}−max{𝐿₀,𝑥₃}=𝑥₁+max{0,𝑥₂−𝐿₀}−max{0,𝑥₃−𝐿₀}. (6.9)
Then, we renormalize them and put 𝑓_𝑖,𝑗∶[0,1]→[0,1] by 𝑓_𝑖,𝑗1(𝑥)=𝐿𝑓_LV(𝑖,𝑗,𝐿𝑥)∈[0,1]. (6.10) Let us divide the interval [0,1] into 𝐿+1 intervals, and denote them as 𝐼₀<𝐼₁<⋯<𝐼_𝐿. Then, 𝑓_𝑖,𝑗(𝐼_𝑙)⊂𝐼_{Φ(𝑙,𝑖,𝑗)} are satisfied for all 𝑖,𝑗,𝑙∈{0,1,…,𝐿}. This is the desired family of maps.
This completes the proof.

6.3. Associated Hypersurfaces

In order to avoid minus sign in the defining equations, one can transpose some terms there. Then, the equation can be expressible by max-plus equations in both sides. So when a bounded automaton is determined by a relative (max,+) function 𝜑, then it associates the pair of (max,+) functions (𝜑₁,𝜑₂).

For example, the LV-CA can be immediately rewritten as 𝑉^𝑡+1_𝑛+1+max{𝐿,𝑉_𝑛^𝑡+1}=𝑉^𝑡_𝑛+1+max{𝐿,𝑉^𝑡_𝑛+2}. (6.11)

Let us consider a bounded automaton defined by a relative (max,+) function: 𝑉^𝑡+1_𝑛+1=𝜑(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1). (6.12) Let 𝜑₁ and 𝜑₂ be two max-plus equations with respect to 𝜑. Then, the above equation can be rewritten as 𝜑₁(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉^𝑡+1_𝑛+1)=𝜑₂(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1),𝑛=0,1,…, (6.13) for some 𝑘 and 𝑙. Notice that 𝜑₁ can be rewritten as 𝜑₁(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉^𝑡+1_𝑛+1)=𝑉^𝑡+1_𝑛+1+𝜑₁(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1) (6.14) for another some (max,+) function 𝜑₁, and they satisfy the relation 𝜑(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1)=𝜑₂(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1)−𝜑₁(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1)=𝜑₂(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉_𝑛^𝑡+1)−𝜑₁(𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉^𝑡+1_𝑛+1)+𝑉^𝑡+1_𝑛+1. (6.15) One may assume that both 𝜑₁ and 𝜑₂ have the same 𝑁 variables.

Let 𝑓¹_𝑡 and 𝑓²_𝑡 be the associated 𝐑 polynomials and assign the complex variables 𝐳=(𝑧₁,𝑧₂,…,𝑧_𝑁+1) with respect to (𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉^𝑡+1_𝑛+1).

Definition 6.3. The associated affine hypersurfaces are a parameterized family of hypersurfaces given by the following equations: 𝑉(𝐴)_𝑡={𝐳∈𝐂^𝑁+1∶𝑓¹_𝑡(𝐳)=𝑓²_𝑡(𝐳)}. (6.16)

Example 6.4. Let us consider the LV cell automaton. Let us put a parameterized polynomial of degree 2 by 𝑓_𝑡(𝑧,𝑤)=𝑡^𝐿𝑧+𝑧𝑤. Then, the corresponding equation becomes 𝑓_𝑡(𝑧₄,𝑧₃)=𝑓_𝑡(𝑧₁,𝑧₂), where each 𝑧_𝑖 corresponds to 𝑧₁↔𝑉^𝑡_𝑛+1, 𝑧₂↔𝑉^𝑡_𝑛+2, 𝑧₃↔𝑉_𝑛^𝑡+1, and 𝑧₄↔𝑉^𝑡+1_𝑛+1. Thus the associated hypersurfaces 𝑉(LV)_𝑡⊂𝐂⁴ are a family defined by 𝑉(LV)_𝑡={(𝑧₁,𝑧₂,𝑧₃,𝑧₄)∈𝐂⁴∶𝑓_𝑡(𝑧₄,𝑧₃)=𝑓_𝑡(𝑧₁,𝑧₂)}. (6.17)

6.3.1. The Associated Complex Dynamics

Let us consider a bounded automaton 𝐴 given by a relative (max,+) function 𝜑. As above, it is given by a pair of (max,+) functions (𝜑₁,𝜑₂). Let 𝑓_𝑡 and (𝑓¹_𝑡,𝑓²_𝑡) be the associated rational and the pair of polynomial functions with respect to 𝜑 and (𝜑₁,𝜑₂) mutually. Notice that they are related in the following way: 𝑓_𝑡(𝑧₁,𝑧₂,…,𝑧_𝑁𝑧)=^𝑁+1𝑓²_𝑡(𝑧₁,𝑧₂,…,𝑧_𝑁)𝑓¹_𝑡(𝑧₁,𝑧₂,…,𝑧_𝑁,𝑧_𝑁+1). (6.18)

For the LV cell automaton, 𝑓_𝑡 above is given by 𝑓_𝑡(𝑧₁,𝑧₂,𝑧₃)=(𝑡^𝐿+𝑧₃)⁻¹(𝑡^𝐿𝑧₁+𝑧₁𝑧₂). (6.19)

Now for 𝑡=0,1,…, let us consider a flow given by infinite sequences of complex numbers (𝑧^𝑡₁,𝑧^𝑡₂,𝑧^𝑡₃,…)∈𝐂^∞ which obey the equation 𝑧^𝑡+1_𝑛+1=𝑓_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1). (6.20) Notice that, if (𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1)∈𝐑^𝑁_>0, then 𝑧^𝑡+1_𝑛+1∈(0,∞).

Since it also satisfies the equation 𝑓¹_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧^𝑡+1_𝑛+1)=𝑓²_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1), (6.21) each building block 𝐳^𝑡_𝑛=(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1) lies on 𝑉(𝐴)_𝑡 for all 𝑛≥𝑘 and 𝑡≥𝑙. We say that the sequences {𝐳^𝑡_𝑛}_{𝑛≥𝑘,𝑡≥𝑙}⊂𝑉(𝐴)_𝑡 (6.22) are the induced dynamics on 𝑉(𝐴)_𝑡.

6.3.2. Stability

So far, we have discussed by fixing the parameter 𝑡. Here we consider the asymptotics of flows as 𝑡→∞.

Let (𝜑₁,𝜑₂) be a bounded automaton given by two max-plus functions with the 𝑁+1 variables (𝑉^𝑡−𝑙_𝑛−𝑘,…,𝑉^𝑡+1_𝑛+1) as above. Let us denote the associated polynomials 𝜑^𝑖_𝑡 and 𝑓^𝑖_𝑡, 𝑖=1,2.

Lemma 6.5. Let 𝑧^𝑡∈𝐑^𝑁+1_>0 be a family of points. Suppose that the convergent condition 𝐱≡lim_𝑡→∞Log_𝑡(𝐳^𝑡) (6.23) is satisfied. Then, 𝑥∈𝐑^𝑁+1 satisfies the following equation: 𝜑₁(𝐱)=𝜑₂(𝐱). (6.24)

Proof. We recall the equality 𝑓_𝑡=log_𝑡⁻¹∘𝜑_𝑡∘Log_𝑡 on the positive real numbers. Thus they satisfy the following equation: 𝜑¹_𝑡∘Log_𝑡(𝐳^𝑡)=𝜑²_𝑡∘Log_𝑡(𝐳^𝑡). (6.25) Then the conclusion holds, since 𝜑^𝑖_𝑡 converge to the original max-plus functions 𝜑^𝑖. This completes the proof.

Let [𝑣]∈𝐙 be the integer part of 𝑣∈𝐑. For a sequence 𝐯=(𝑣₁,𝑣₂,…), we denote by [𝐯]≡([𝑣₁],[𝑣₂],…).

Definition 6.6. Let 𝐴 be a bounded automaton, and choose parameterized families of sequences {𝐳^𝑡_𝑛}_{𝑛≥𝑘,𝑡≥𝑙}⊂𝐂^𝑁+1.
We say that they are stable, if there is 𝑡₀ so that, for all 𝑡≥𝑡₀, {[Log_𝑡(𝐳^𝑡_𝑛)]}_{𝑛≥𝑘,𝑡≥𝑙}⊂𝐙^𝑁+1 (6.26) gives a flow of the solutions of the original automaton 𝐴.

Let 𝐴 be the LV cell automaton. Then, as in Section 6.2, there is a family of interval maps {𝑓_𝑖,j}_𝑖,𝑗 so that the flow of the solutions of LV-CA can be represented by interactions of maps between {𝑓_𝑖,𝑗}. Thus existence of stable families above will be heavily influenced by stability of dynamical properties of the family {𝑓_𝑖,𝑗} under continuous deformations of these maps.

6.4. Cell Automatic Varieties

Let 𝐴₁ and 𝐴₂ be two cell automata, and consider an integer-valued flow {𝐕^𝑡=(𝑉^𝑡₁,𝑉^𝑡₂,…)}_𝑡≥0⊂𝐙^∞. (6.27) If it satisfies both equations for 𝐴₁ and 𝐴₂, then we say that {𝐕^𝑡}_𝑡 is a flow for 𝐴₁ and 𝐴₂.

Let us denote the set of such flows by 𝐅(𝐴₁,𝐴₂)={{𝐕^𝑡}_{𝑡=0,1,2,…}⊂𝐙^∞∶flowsforboth𝐴₁,𝐴₂}. (6.28) We say that 𝐅(𝐴₁,𝐴₂) is a cell automatic variety for 𝐴₁ and 𝐴₂.

Let 𝐴₁,…,𝐴_𝑚 be a family of cell automata. By considering flows for them, one also obtains the cell automatic variety 𝐅(𝐴₁,…,𝐴_𝑚).

6.4.1. Compatible Automata

Let 𝐴₁ and 𝐴₂ be two cell automata given by 𝜑 and 𝜑^′, respectively: 𝑉^𝑡+1_𝑛+1=𝜑(^′)({𝑉^𝑡_𝑛−𝑘,…,𝑉^𝑡_𝑛+𝑙},{𝑉^𝑡+1_{𝑛−𝑘′},…,𝑉_𝑛^𝑡+1})[∗](^′). (6.29) Let us denote the following sets: 𝐕^𝑡_𝑛≡(𝑉^𝑡_𝑛−𝑘,…,𝑉^𝑡_𝑛+𝑙,𝑉^𝑡+1_{𝑛−𝑘′},…,𝑉_𝑛^𝑡+1)∈𝐙^𝑁. (6.30)

We say that 𝐴₂ is compatible with 𝐴₁ if the following holds; suppose that 𝑉^𝑡_𝑛 satisfies both [∗] and [∗]^′. Let 𝑉^𝑡_𝑛+1 be determined by [∗]^′. Then, it also satisfies [∗].

In the case of compatible automata (𝐴₁,𝐴₂), the cell automatic variety 𝐅(𝐴₁,𝐴₂) will be nonempty, which does not hold in general.

Example 6.7. Let 𝐴₁ and 𝐴₂ be both automata given by linear maps as follows: 𝑢_𝑛+1=𝛼𝑢_𝑛+𝛽𝑢_𝑛−1+𝛾,(𝐴₁𝑢),_𝑛−1=𝑎𝑢_𝑛+𝑏,(𝐴₂). (6.31) If these coefficients satisfy the following relations: (1−𝛽)𝑎=𝛼,(1−𝛽)𝑏=𝛾, (6.32) then 𝐴₂ is compatible with 𝐴₁.

6.4.2. Associated Varieties

Let 𝐴₁ and 𝐴₂ be two CA. Let us denote the associated 𝐑-polynomials by 𝑓^𝑖_𝑡 and 𝑔^𝑖_𝑡 for 𝑖=1,2, respectively. Then, one obtains the parameterized affine algebraic varieties given by the following equations: 𝑉(𝐴₁,𝐴₂)_𝑡={𝐳∈𝐂^𝑁∶𝑓¹_𝑡(𝐳)=𝑓²_𝑡(𝐳),𝑔¹_𝑡(𝐳)=𝑔²_𝑡(𝐳)}. (6.33)

Let us consider the associated systems of complex dynamics. Let (𝑧^𝑡₁,𝑧^𝑡₂,𝑧^𝑡₃,…)∈𝐂^∞ which obey the following system of the equations: 𝑓¹_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧^𝑡+1_𝑛+1)=𝑓²_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1𝑔),_1𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧^𝑡+1_𝑛+1)=𝑔²_𝑡(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1). (6.34) Again each building block 𝐳^𝑡_𝑛=(𝑧^𝑡−𝑙_𝑛−𝑘,…,𝑧_𝑛^𝑡+1) lies on 𝑉(𝐴₁,𝐴₂)_𝑡, and we say that {𝐳^𝑡_𝑛}_𝑛,𝑡 is the induced dynamics on 𝑉(𝐴₁,𝐴₂)_𝑡.

Let 𝐴₁,…,𝐴_𝑚 be a family of CA. Then, by the same way as above, one obtains the following parameterized associated affine algebraic variety: 𝑉(𝐴₁,…,𝐴_𝑚)_𝑡={𝐳∈𝐂^𝑁∶(𝑓^𝑘_𝑡)¹(𝐳)=(𝑓^𝑘_𝑡)²(𝐳),𝑘=1,…,𝑚}, (6.35) where (𝑓^𝑘_𝑡)^𝑖, 𝑖=1,2, are the associated 𝐑-polynomials with respect to 𝐴_𝑘.

6.5. Duality

Here we introduce a new duality on the set of automata. It passes through the projective duality between algebraic varieties [17].

Let 𝑉 be a complex 𝑛-dimensional vector space and let 𝑃(𝑉) be its projective space. There is a natural isomorphism 𝑃(𝑉)≅𝑃^∗(𝑉^∗), where 𝑃^∗(𝑊) is the set of all hyperplanes in 𝑊 and 𝑉^∗ is the dual space to 𝑉.

Projective duality generalizes this operation to varieties. Let us quickly describe its construction. Let 𝑋⊂𝑃(𝑉) be an algebraic variety. Then, one can associate another variety 𝑋^∨⊂𝑃(𝑉^∗) as follows. A hyperplane 𝐻⊂𝑃(𝑉) is said to be tangent to 𝑋 if there exists a smooth point 𝑥∈𝐻∩𝑋 and the tangent space of 𝑋 at 𝑥 is contained in 𝐻. Let 𝑋^∗⊂𝑃^∗(𝑉) be all of the set of tangent hyperplanes, and passing through the above isomorphism, one obtains a set 𝑋^∨⊂𝑃(𝑉^∗) which is the desired one. It is called the projective dual variety. In the case when 𝑋^∨⊂𝑃(𝑉^∗) is a hypersurface, then its defining polynomial Δ_𝑋 is called the 𝑋-discriminant.

Let {𝐴₁,…,𝐴_𝑚} be a family of automata. Thus one obtains a parameterized family of projective varieties 𝑉({𝐴_𝑖})_𝑡⊂𝐂𝐏^𝑁 by taking closure of the associated affine varieties.

Let 𝑉({𝐴_𝑖})^∨_𝑡⊂𝐂𝐏^𝑁 be the corresponding parameterized projective dual varieties.

Suppose that these are hypersurfaces, and denote the defining functions by Σ_𝑗𝑡^𝛼𝑗𝑎^𝑗𝑧^𝑗. When one can modify the polynomial as Σ_𝑗𝑡^𝛼𝑗𝑤^𝑗 by change of variables 𝑎_𝑙𝑧_𝑙=𝑤_𝑙, then we call Σ_𝑗𝑡^𝛼𝑗𝑤^𝑗 the {𝐴_𝑖}-discriminant and denote it by Δ({𝐴_𝑖})_𝑡. (6.36) We will denote the corresponding modified varieties by 𝑉({𝐴_𝑖})^∨_𝑡 (6.37) and call them as the associated dual varieties with respect to {𝐴_𝑖}_𝑖. They are isomorphic with 𝑉({𝐴_𝑖})^∨_𝑡.

Definition 6.8. Let 𝜑_𝑡 be the associated 𝑅_𝑡-polynomials to Δ({𝐴_𝑖})_𝑡). The automaton defined by 𝜑_∞=lim_𝑡→∞𝜑_𝑡 is called the projectively dual automaton.

We denote the projectively dual automaton by {𝐴_𝑖}^∨. (6.38)

6.5.1. Curves in 𝐂𝐏²

In general it is not easy to calculate the defining equations of the projective varieties. However as far as the simplest case is that of the curves in 𝐂𝐏², one can write down them explicitly. Using this fact, we verify the following.

Proposition 6.9. Consider [max{𝑎𝑢_𝑛,𝛼+𝑎𝑢_𝑛+1}=𝑐]^∨𝑎=max{𝛼𝑎−1(𝑐−𝑎𝑎)+𝑢𝑎−1^𝑛+1,𝑎𝑐+𝑎𝑎−1𝑢𝑎−1^𝑛}=𝑐. (6.39)

Proof. Let 𝑋⊂𝐂𝐏² be an irreducible curve. Then, it is known that 𝑋^∨⊂𝐂𝐏² is also another irreducible one. In the affine coordinate, if 𝑋 has a parameterization 𝑥=𝑥(𝑠) and 𝑦=𝑦(𝑠), 𝑠∈𝐶, then 𝑋^∨ has a parameterization given by the following (see [17]): 𝑝(𝑠)=−𝑦^′(𝑠)𝑥^′(𝑠)𝑦(𝑠)−𝑥(𝑠)𝑦^′𝑥(𝑠),𝑞(𝑠)=^′(𝑠)𝑥^′(𝑠)𝑦(𝑠)−𝑥(𝑠)𝑦^′.(𝑠) (6.40)
Using this, let us consider the very simple case above. Let 𝑎≥2, let 𝛼 and 𝑐 be integers, and consider an automaton 𝐴 given by max{𝑎𝑢_𝑛,𝛼+𝑎𝑢_𝑛+1}=𝑐. (6.41) The associated polynomial and the associated varieties are given by 𝑋={(𝑥,𝑦)∈𝐂²⊂𝐂𝐏²∶𝑥^𝑎+𝑡^𝛼𝑦^𝑎=𝑡^𝑐}. (6.42) Choosing a parameterization as 𝑥=𝑠,𝑦=𝑡^−𝛼/𝑎(𝑡^𝑐−𝑠^𝑎)^1/𝑎, (6.43) one can immediately obtain the parameterization of 𝑋^∨ as 𝑡^{(𝑎/(𝑎−1))(𝑐−𝛼/𝑎)}𝑞^{𝑎/(𝑎−1)}+𝑡^{𝑎𝑐/(𝑎−1)}𝑝^{𝑎/(𝑎−1)}=𝑡^𝑐(∗) (6.44) which gives the dual varieties 𝑉(𝐴)^∨_𝑡={(𝑝,𝑞)∈𝐂²⊂𝐂𝐏²∶(∗)}. (6.45) Thus the projectively dual automaton admits the desired presentation. This completes the proof.

6.6. Transformations

Let us take two automata 𝑂 and 𝑂^′, and let 𝑂={𝑂=𝑂₀→𝑂₁→⋯𝑂_𝑚=𝑂^′}∈Ω(𝑂,𝑂^′) be an invertible path, as in Section 4.3, whose associated 𝐑-polynomials are given as the pairs (𝑓¹_𝑡,𝑓²_𝑡)_𝑖, 𝑖=0,…,𝑚. Then, correspondingly the associated hypersurfaces {𝑉(𝑂_𝑖)}_{𝑖=0,…,𝑚}⊂𝐂^𝑁 are obtained. We call this a lifting of 𝑂∈Ω_𝑂.

Question 3. What are geometric structure of such liftings?

Recall that we have obtained other cell automata 𝑂₁ and 𝑂₂ during the process of transforming from BBS to LV; see Section 4.1. Similarly, as above we rewrite these as 𝑈^𝑡+1_𝑛+1+max{1,𝑈^𝑡_𝑛+1}=𝑈^𝑡_𝑛+max{1,𝑈_𝑛^𝑡+1𝑆},_𝑡+1𝑛+1+max{𝑆_𝑛^𝑡+1+1,𝑆^𝑡_𝑛+1}=𝑆^𝑡_𝑛+𝑆_𝑛^𝑡+1+1. (6.46) Let us assign the variables as 𝑧₁⟷𝑋^𝑡_𝑛,𝑧₂⟷𝑋^𝑡_𝑛+1,𝑧₃⟷𝑋_𝑛^𝑡+1,𝑧₄⟷𝑋^𝑡+1_𝑛+1 (6.47) for both 𝑋=𝑆 and 𝑋=𝑈. Then, we have the corresponding polynomial pairs as 𝑓¹_𝑡(𝑧₂,𝑧₄)=𝑧₄(𝑡+𝑧₂),𝑓²_𝑡(𝑧₁,𝑧₃)=𝑧₁(𝑡+𝑧₃𝑓),for𝑈,_1𝑡(𝑧₂,𝑧₃,𝑧₄)=𝑧₄(𝑡𝑧₃+𝑧₂),𝑓²_𝑡(𝑧₁,𝑧₃)=𝑡𝑧₃𝑧₁,for𝑆. (6.48) Thus one has obtained three associated hypersurfaces as follows: 𝑉(LV),𝑉(𝑈),𝑉(𝑆)⊂𝐂⁴. (6.49) Below we study one aspect of Question 3 for this case.

6.6.1. Representatives by Homogeneous Maps

Let us consider a bounded automaton 𝐴 given by a relative (max,+) function 𝜑. As in Section 6.3.1, it is given by a pair of (max,+) functions (𝜑₁,𝜑₂). Let (𝑓¹_𝑡,𝑓²_𝑡) be the associated polynomials with respect to (𝜑₁,𝜑₂).

If both 𝑓¹_𝑡 and 𝑓²_𝑡 are homogeneous polynomials of the same degree, then we say that 𝐴 is a homogeneous cell automaton.

The LV-CA is not homogeneous; see Section 6.3. However one can deform it so that the result becomes homogeneous as below.

Let 𝑂 be a cell automaton. We say that 𝑂 admits a homogeneous representative if there is an invertible transformation 𝑂=𝑂₀→𝑂₁→⋯→𝑂_𝑘 so that 𝑂_𝑘 is homogeneous.

Lemma 6.10. LV cell automaton admits a homogeneous representative.

Proof. Let 𝑂₀=LV→𝑂₁→𝑂₂→𝑂₃=BBS be the invertible path in Lemma 4.1. Then, certainly 𝑂₂ is homogeneous as above.
This completes the proof.

Let 𝑂 be a homogeneous cell automaton. Then, one obtains the associated projective hypersurface 𝑉(𝑂)⊂𝐂𝐏^𝑁−1. (6.50) We say that 𝑉(𝑂) is the associated projective hypersurface.

7. Interaction Graphs

Let 𝑓,𝑔∶[0,1]→[0,1] be two interval maps and let Φ(𝑥,𝑓,𝑔)∶𝑋₂→𝑋₂ be the interaction map. Recall that 𝜋∶[0,1]⧵1/2→{0,1} is the projection. Let us choose another map 𝑑∶[0,1]→[0,1].

Suppose that, for a point 𝑧∈[0,1] and some 𝑘∈𝑋₂, the following equality holds: Φ(𝑥,𝑓,𝑔)(𝑘)=𝜋((𝑑(𝑧),𝑑²(𝑧),…))≡(𝜋(𝑑(𝑧)),𝜋(𝑑²(𝑧)),…). (7.1) Then, we express this by a marked oriented edge as (𝑓,𝑥)−−−−→^(𝑔,𝑘)(𝑑,𝑧). (7.2)

Let us choose families of maps {𝑓₀,…,𝑓_𝑘} and points {𝑥₀,…,𝑥_𝑙}. For each (𝑖,𝑗,𝑥)∈{0,…,𝑘}²×{𝑥₀,…,𝑥_𝑙}, let us assign an element 𝑘(𝑖,𝑗,𝑥)∈𝑋₂. Thus we obtain another family {𝑘(𝑖,𝑗,𝑥_ℎ)}^{𝑖,𝑗=𝑘,ℎ=𝑙}_{𝑖,𝑗,ℎ=0}⊂𝑋₂. Then, we put the following two sets:𝑉={(𝑓_𝑖,𝑥_𝑗)∶0≤𝑖≤𝑘,0≤𝑗≤𝑙}(thesetofvertices),𝐸={𝑒_{𝑖,𝑗,𝑘}∶(𝑓_𝑖,𝑥_ℎ)−−−−−−−−−−→^{(𝑓𝑗,𝑘(𝑖,𝑗,𝑥ℎ))}(𝑓_𝑘,𝑥_𝑣)∶}(thesetofedges). (7.3)

Definition 7.1. An interaction graph is a marked oriented graph, where the sets of vertices 𝑉 and edges 𝐸 are given as above. We denote it by 𝐺({𝑓_𝑖}^𝑘_𝑖;{𝑥_𝑗}^𝑙_𝑗;{𝑘(𝑖,𝑗,𝑥_ℎ)}^{𝑖,𝑗=𝑘,ℎ=𝑙}_{𝑖,𝑗,ℎ=0}). (7.4) We will denote the set of interaction graphs arising from {𝑓₀,…,𝑓_𝑘} and {𝑥₀,…,𝑥_𝑙} by 𝔊({𝑓_𝑖}^𝑘_𝑖=0;{𝑥_𝑗}^𝑙_𝑗=0). (7.5) Notice that this is a finite set.

Let us put 𝑋₂^𝑘,𝑙≡𝑋^𝑘2₂^+𝑙=𝑋₂×𝑋₂×⋯×𝑋₂. (7.6) Then, any element in 𝑋₂^𝑘,𝑙 can be written as 𝑘(𝑖,𝑗,𝑥) as above. Then, the family of the interaction map gives a map Φ∶𝑋₂^𝑘,𝑙→𝑋₂^𝑘,𝑙, (7.7) where Φ({𝑘(𝑖,𝑗,𝑥)})={𝑘^′(𝑖,𝑗,𝑥)},𝑘^′(𝑖,𝑗,𝑥)≡Φ(𝑓_𝑖,𝑓_𝑗,𝑥)(𝑘(𝑖,𝑗,𝑥)). (7.8)

This induces a map on the set of the interaction graph as Φ_∗∶𝔊({𝑓_𝑖}^𝑘_𝑖=0;{𝑥_𝑗}^𝑙_𝑗=0)→𝔊({𝑓_𝑖}^𝑘_𝑖=0;{𝑥_𝑗}^𝑙_𝑗=0) (7.9) byΦ_∗(𝐺({𝑓_𝑖}^𝑘_𝑖;{𝑥_𝑗}^𝑙_𝑗;{𝑘(𝑖,𝑗,𝑥_ℎ)}))=𝐺({𝑓_𝑖}^𝑘_𝑖;{𝑥_𝑗}^𝑙_𝑗;Φ({𝑘(𝑖,𝑗,𝑥_ℎ)})). (7.10) Thus one obtains a sequence of the interaction graphs as follows:(𝐺₀,𝐺₁𝐺,…),_𝑖=𝐺({𝑓_𝑖}^𝑘_𝑖;{𝑥_𝑗}^𝑙_𝑗;Φ^𝑖({𝑘(𝑖,𝑗,𝑥_ℎ)}^{𝑖,𝑗=𝑘,ℎ=𝑙}_{𝑖,𝑗,ℎ=0})). (7.11) This gives a dynamics of the interaction graphs. Below we will formulate several geometric spaces arising from dynamics of the interaction graphs which we call the spaces from the interaction graphs.

7.1. Veronese Map

Here we have an easy example of spaces from the interaction graphs. Let ({𝑓_𝑖},{𝑥_𝑗},{𝑘(𝑖,𝑗,𝑥_ℎ)}) be an interaction system, and denote the corresponding interaction graphs by (𝐺₀,𝐺₁,…). Passing through the forgetful map, one obtains a sequence of finite graphs (𝐺^′₀,𝐺^′₁,…).

For each vertex 𝑣∈𝐺^′_𝑖, let 𝑒(𝑣) be the number of the edges with a common vertex 𝑣. Let us fix 𝑚≥0 and put 𝑃(𝑚,𝐺)={(𝑖₀,…,𝑖_𝑘)∶Σ^𝑘_𝑎=0𝑒(𝑖_𝑎)=𝑚,𝑘≤𝑁}. (7.12)

Let 𝑁+1 be the number of the edges in the interaction graphs. The Veronese map with respect to {𝐺^′_𝑖}_𝑖 is a family of embeddings:𝐼_𝑖∶𝐂𝐏^𝑁↪𝐂𝐏^𝑀,[𝑥₀,…,𝑥_𝑁]→[{𝑥^𝑖0₀…,𝑥^𝑖𝑘_𝑘∶(𝑖₀,…,𝑖_𝑘)∈𝑃(𝑚,𝐺_𝑖)}], (7.13) where 𝑁_𝑖=#𝑃(𝑚_𝑖,𝐺_𝑖)+1 and 𝑀=(𝑁+1)𝑁/2 is the maximum number of the edges in the graphs.

7.1.1. Reduction to Dynamics of Toric Ideals

Let 𝔊 be the set of finite graphs, and let 𝐹∶𝔊({𝑓_𝑖}^𝑘_𝑖=0;{𝑥_𝑗}^𝑙_𝑗=0)→𝔊 (7.14) be the forgetful map. Then, one obtains a family of finite graphs 𝐺₁,𝐺₂,…⊂𝔊 (7.15) consisted by the images of 𝐹 of the interaction graphs. We call them just the associated graphs.

Let {𝐺₀,𝐺₁,…} be a family of finite graphs. We say that the family is strongly regular if the numbers of edges of 𝐺_𝑖 are all the same. In Section 7.1, we will always assume that sequences consisted by the images of 𝐹 of the interaction graphs are strongly regular.

For each 𝐺∈𝔊, let us associate a configuration 𝔄={𝑎₁,…,𝑎_𝑚}⊂𝐙^𝑁 (7.16) as follows, where 𝑁=(𝑘+1)(𝑙+1) is the number of the vertices. Let us make a numbering of the set of vertices, 𝑣₁,…,𝑣_𝑁, and let 𝑒_𝑖∈𝐙^𝑁 be the unit vector (0,…,0,1,0,…,0), where 1 appears only at the 𝑖th. Then, 𝑒_𝑖+𝑒_𝑗∈𝔄 if and only if 𝑣_𝑖 and 𝑣_𝑗 are mutually connected by an edge.

Thus one obtains a reduction from dynamics of the interaction graphs to the one of the following configurations: {𝔄₀,𝔄₁,…}, (7.17) which we call the transcripted configurations.

For a configuration 𝔄={𝑎₁,…,𝑎_𝑚}⊂𝐙^𝑁, we associate a Laurent polynomial: 𝐂[𝔄]≡𝐂[𝑡^𝑎1,…,𝑡^𝑎𝑚]⊂𝐂[𝑡₁^±1,…,𝑡_𝑁^±1], (7.18) where 𝑡^𝑎𝑘=𝑡^𝑎1𝑘₁⋯𝑡^𝑎𝑁𝑘_𝑁, 𝑎_𝑘=(𝑎¹_𝑘,…,𝑎^𝑁_𝑘). We call 𝐂[𝔄] as the associated toric ring.

By assigning 𝑦_𝑖→𝑡^𝑎𝑖, one obtains a ring homomorphism 𝜋∶𝐂[𝑦₁,…,𝑦_𝑚]→𝐂[𝔄] (7.19) and its kernel 𝐼_𝔄 is called the toric ideal.

For each interaction graph 𝐺_𝑖∈𝔊({𝑓_𝑖}^𝑘_𝑖=0,{𝑥_𝑗}^𝑙_𝑗=0), one forgets markings on edges and orientation and then obtains a finite graph 𝐹(𝐺_𝑖). Then, one can assign a configuration 𝔄_𝑖⊂𝐙^𝑁. Now correspondingly one has the associated toric ideal 𝐼_𝑖⊂𝐂[𝑦₁,…,𝑦_𝑚]. Thus one has obtained a sequence of toric ideals as 𝐼₀,𝐼₁,…,𝐼_𝑘,𝐼_𝑘+1,…⊂𝐂[𝑦₁,…,𝑦_𝑚] (7.20) which we will call the associated ideals. Notice that there are a finite number of ideals {𝐽₁,…,𝐽_𝑑} such that each 𝐼_i coincides with one of {𝐽_𝑗}. We say that the associated ideals are regular if all 𝐼_𝑖 has the same dimension.

Let us fix a total ordering < on the set of monomials of the polynomial ring 𝐂[𝑦₁,…,𝑦_𝑚], for example, lexicographic or its reverse ones.

Let 𝐼⊂𝐂[𝑦₁,…,𝑦_𝑚] be an ideal, and let 𝑧={𝑧₁,…,𝑧_𝑘} be a generating set. We denote by int𝑧_𝑖 to imply the leading term of the polynomial with respect to the ordering. Let int 𝐼⊂𝐂[𝑦₁,…,𝑦_𝑁] be another ideal generated by int𝑧 for all 𝑧∈𝐼. It is called the initial ideal. We say that 𝑧 is a Gröbner basis if int𝐼=gen{int𝑧₁,…,int𝑧_𝑘}. (7.21)

Notice that the sequence of the toric ideals {𝐼_𝑖}_𝑖 is obtained originally from the data ({𝑓_𝑖}_𝑖,{𝑥_𝑗}_𝑗,{𝑘(𝑖,𝑗,𝑥_ℎ)}). The following problem seems natural, since, in some cases, these interactions come from some combinatoric structures, like cell automata [3].

Question 4. Can one find an algorithm to find out Gröbner basis successively for 𝐼₀,𝐼₁,… from the interaction data?

7.1.2. Gröbner Fans and Translated Varieties

Let 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐑^𝑚 be a vector which is called the weight vector. Then, the weight of a monomial 𝑦^𝑎1₁⋯𝑦^𝑎𝑚_𝑚 is equal to ⟨𝑤,𝑎⟩=𝑤₁𝑎₁+𝑤₂𝑎₂+⋯+𝑤_𝑚𝑎_𝑚. For each polynomial 𝑓=Σ𝑐_𝑎𝑦^𝑎∈𝐂[𝑦₁,…,𝑦_𝑚], let in_𝑤(𝑓)=Σ_𝑏𝑐_𝑏𝑦^𝑏, where any 𝑏 satisfies ⟨𝑤,𝑏⟩=max_𝑐𝑎≠0⟨𝑤,𝑎⟩.

Let 𝐼 be an ideal. Then, we denote the corresponding initial ideal in_𝑤(𝐼) generated by all elements of the form in_𝑤(𝑓). It is known that, for every ideal 𝐼 and ordering <, a weight vector 𝑤 is associated which satisfies in_<(𝐼)=in_𝑤(𝐼).

For an ideal 𝐼, we say that 𝑤 and 𝑤^′ are equivalent if they give the same initial ideal int_𝑤𝐼=int_𝑤′𝐼. The equivalence class ℭ(𝐼,𝑤)={𝑤^′∈𝐑^𝑚∶int_𝑤𝐼=int_𝑤′𝐼} (7.22) is an open polyhedral cone in 𝐑^𝑚.

The set of the cones {ℭ(𝐼,𝑤)}_𝑤 is finite and defines a polyhedral fan 𝔉(𝐼). We say that it is the Gröbner fan of 𝐼 (see [18]).

For each fan, there associated with a toric variety. Thus for each interaction graph 𝐺, one associates with a toric variety 𝑋_𝐺 which we call the translated variety.

Thus corresponding to a sequence of the associated ideals 𝐼₀,𝐼₁,…, one obtains a sequence of fans over 𝐑^𝑚 and the associated translated varieties 𝑋=(𝑋₀,𝑋₁,…,𝑋_𝑖,…). (7.23)

Remark 7.2. In order to study structure of 𝑋, one may use resultants for the defining polynomials of these ideals.
Now we have started from a finite data 𝐃=({𝑓_𝑖}^𝑘_𝑖=0,{𝑥_𝑗}^𝑙_𝑗=0,{𝑎(𝑖,𝑗,ℎ)}^{𝑖,𝑗=𝑘,ℎ=𝑙}_{𝑖,𝑗,ℎ=0}). (7.24) We will call such data an interaction data.
Then, we have obtained a sequence of the interaction graphs: 𝐺₀,𝐺₁𝐺,…,_𝑖=𝐺({𝑓_𝑖}^𝑘_𝑖=0,{𝑥_𝑗}^𝑙_𝑗=0,Φ^𝑖_∗({𝑎(𝑖,𝑗,ℎ)})). (7.25) By forgetting extra data, one obtains a sequence of the transcripted configurations: 𝔄₀,𝔄₁,…, (7.26) where 𝔄_𝑖⊂𝐙^𝑁. Then, one has obtained a sequence of the toric ideals {𝐼₀,𝐼₁,…} and the translated toric varieties: {𝑋₀,𝑋₁,…}⊂𝐂𝐏^𝑚−1, (7.27) where each 𝑋_𝑖 coincides with one in a finite set of toric varieties {𝑌₁,…,𝑌_𝑙}⊂𝐂𝐏^𝑚−1.

7.1.3. Correspondence on Polytopes

Let ({𝑓_𝑖}^𝑘,{𝑥_𝑗}^𝑙,{𝑎(𝑖,𝑗,ℎ)}) be a triple, and consider the corresponding interaction graphs {𝐺_𝑖}^∞_𝑖=0. Then, there are finite configurations {𝔄₁,…,𝔄_𝑙}⊂𝐙^𝑁 such that each 𝐺_𝑖 associates one of 𝔄_𝑗 for some 𝑗=𝑗(𝑖).

Let 𝔄⊂𝐙^𝑁 be a configuration, and let 𝑚 be the number of the elements in 𝔄.

Let Δ be a triangulation of 𝔄. Then, every 𝜓∈𝐑^𝑚 defines the corresponding piecewise linear function 𝑔_𝜓,Δ on 𝐑^𝑚 satisfying 𝑔_𝜓,Δ(𝑎_𝑖)=𝜓(𝑎_𝑖) for each vertex 𝑎_𝑖 of Δ which is affine on each simplex of Δ.

Then, we put ℭ(𝔄,Δ)≡{𝜓∈𝑅^𝑚∶𝑔_𝜓,Δisconcaveand𝑔_𝜓,Δ≥𝜓(𝑎_𝑖)whenever𝑎_𝑖isnotavertexofΔ}. (7.28) Moreover, ℭ(𝔄,Δ) is a closed polyhedral cone, and 𝔉(𝔄)≡{ℭ(𝔄,Δ)∶Δisatriangulation} (7.29) forms a complete fan on 𝐑^𝑚 and is called the secondary fan of 𝔄 [17, page 219]. It is known that the Gröbner fan is a refinement of the secondary fan [18].

Let 𝑄⊂𝐑^𝑚 be a polytope. Then, for each 𝑝∈𝑄, let us define the normal cone 𝑁(𝑄,𝑝)≡{𝑣∈𝐑^𝑚∶⟨𝑣,𝑝⟩≥⟨𝑣,𝑦⟩∀𝑦∈𝑄}. (7.30) The set of the normal cones 𝑁(𝑄)=_{𝑝∈Vert𝑄}𝑁(𝑄,𝑝) (7.31) is called the normal fan.

Theorem 7.3 (see [17]). For a configuration 𝔄, there is a polytope 𝑄≡Σ(𝔄) and an assignment of a vertex 𝜑_Δ∈𝑄 for each triangulation Δ such that the normal cone 𝑁(Σ(𝔄),𝜑_Δ) coincides with ℭ(𝔄,Δ).
In particular there is a natural correspondence from the secondary fan 𝔉(𝔄) to the corresponding normal fan 𝑁(𝑄).

Σ(𝔄) is called the secondary polytope.

Let us denote 𝔄={𝑎₁,…,𝑎_𝑚}. Then, one can describe 𝑄 explicitly as follows. Let us put 𝜙_Δ=(𝜙¹_𝛿,…,𝜙^𝑚_Δ)∈𝐑^𝑚,𝜙^𝑖_Δ=Σ{vol(𝜏)∶𝜏∈Δ,𝑎_𝑖∈𝜏}. (7.32) Then, 𝑄=Σ(𝔄) is given by 𝑄≡Conv{𝜙_Δ∶Δisatriangulationof𝔄}, (7.33) where vol(𝜏) is the volume of 𝜏.

Now let {𝔄₀,…,𝔄_𝑙,…}⊂𝐙^𝑁 be the transcripted configurations. Then, correspondingly one obtains other sequences of polytopes: {Σ(𝔄₀),Σ(𝔄₁),…}, (7.34) which is called the secondary transcripted polytopes.

7.2. Dynamics over Local Charts

Let us fix two sets {𝑓_𝑖}^𝑘_𝑖=0 and {𝑥_𝑗}^𝑙_𝑗=0. Then, there associates with finite numbers of toric ideals {𝐽₁,…,𝐽_𝑚}⊂𝐂[𝑦₁,…,𝑦_𝑁].

If we choose a finite subset {𝑎(𝑖,𝑗,ℎ)}⊂𝑋₂, then one obtains an infinite sequence of ideals 𝐼({𝑎(𝑖,𝑗,ℎ)})=(𝐼₀,𝐼₁,…,𝐼_𝑖,…) (7.35) among a finite set {𝐽_𝑗}_𝑗 above.

Let us put all of the set of sequences 𝐈={𝐼({𝑎(𝑖,𝑗,ℎ)})∶{𝑎(𝑖,𝑗,ℎ)}⊂𝑋₂}. (7.36) We call it the sequence of local charts.

One may consider this as though it might be a symbolic dynamics of some “Markov partition’’ over some algebraic variety 𝑉, by regarding each 𝐼_𝑘 as a defining ideal of a local chart of 𝑉.

7.2.1. Algebraic Markov Partition

Let 𝑉 be an algebraic variety with the affine coordinates 𝑉_𝑖 defined by an ideal 𝐽_𝑖. Let us take an automorphism 𝐴 on 𝑉. We say that an affine coordinate {(𝑉_𝑖,𝐽_𝑖)}^𝑚_𝑖=1 is an (stable) algebraic Markov partition for 𝐴 if, for each 𝑖, there is some 𝑗 so that 𝐴(𝑉_𝑖)⊂𝑉_𝑗 (7.37) holds.

Example 7.4. Let 𝐴 be an automorphism on 𝐂𝐏^𝑁 by [𝑧₀,𝑧₁,…,𝑧_𝑁]→[𝑧₁,𝑧₀,𝑧₂,𝑧₃,…,𝑧_𝑁]. Then, for 𝑉_𝑖={[𝑧₀,𝑧₁,…,𝑧_𝑁]∶𝑧_𝑖≠0}, 𝐂𝐏^𝑁 admits an affine covering by 𝑉₀∪𝑉₁. Moreover 𝐴(𝑉_𝑖)=𝑉_𝑖+1 mod 2.
Let 𝐴∶𝑉≅𝑉 be an automorphism, and consider its iteration 𝐴^𝑡∶𝑉≅𝑉. Let us have an algebraic Markov partition by the set {𝑉_𝑖}_𝑖. Then, one obtains a symbolic dynamics of the algebraic Markov partition: Σ(𝐴;{𝑉_𝑙}_𝑙)={(𝑎₀,𝑎₁,…)∶𝐴(𝑉_𝑎𝑖)⊂𝑉_𝑎𝑖+1)}⊂𝑋_𝑚, (7.38) where 𝑚 is the number of the local charts. It can be expressed by a set of sequences of the defining ideals {(𝐼₀,𝐼₁,…)}.
One can consider its converse. Let us put the set of all sequences coming from the algebraic Markov partition: 𝐀={(𝐼₀,𝐼₁,…)}. (7.39)
When one is given an interaction data ({𝑓_𝑖}_𝑖,{𝑥_𝑗}_𝑗,{𝑎(𝑖,𝑗,ℎ)}), then one obtains a sequence of ideals. Thus a fundamental question in symbolic dynamics of ideals will be to construct correspondence from interaction data to algebraic Markov partitions.

Question 5. Let 𝐈 be a set of sequences of ideals among a finite set of ideals. Then, can one construct an algebraic Markov partition {𝐽₁,…,𝐽_𝑘} for an algebraic variety 𝑉 and an automorphism 𝐴, so that 𝐈⊂𝐀 might hold? Namely, when are a set of sequences of ideals symbolic dynamics of algebraic Markov partitions?

Conversely, given 𝐀, can one find some {𝑓_𝑖}^𝑘_𝑖=0 and {𝑥_𝑗}^𝑙_𝑗=0 so that the corresponding 𝐈 might satisfy 𝐈⊂𝐀?

We say that associated ideals 𝐼=(𝐼₀,𝐼₁,…) are regular if they have the same dimension as the others. Let us put ⋃_𝑖𝐼_𝑖={𝐽₁,…,𝐽_𝑘}.

Definition 7.5. Let ({𝑓_𝑖}_𝑖,{𝑥_𝑗}_𝑗,{𝑎(𝑖,𝑗,ℎ)}) be an interaction data, and suppose that the associated ideals by 𝐼=(𝐼₀,𝐼₁,…) are regular. The sequence is called a symbolic flow of an automorphism if there is an algebraic Markov partition for (𝑉,𝐴) with an affine coordinate {(𝑉_𝑖,𝐽_𝑖}^𝑘_𝑖=1 and some 𝑥∈𝑉 so that its orbit {𝐴^𝑛(𝑥)}_𝑛=0,1,… corresponds to the sequence.

We call such pair (𝑉,𝐴) corresponding to 𝐼 a prohedron. (One may imagine as though it represents some state of a protein.)

Let us consider the simplest case. Let us consider the partition of 𝐂𝐏^𝑁=𝑉₀∪𝑉₁ and the involution 𝐴 in Example 7.4.

Lemma 7.6. Let ({𝑓_𝑖}^𝑘_𝑖=0,{𝑥_𝑗}^𝑙_𝑗=0,{𝑎(𝑖,𝑗,ℎ)}) be an interaction data such that the corresponding sequence of the interaction graphs 𝐺₀,𝐺₁,… satisfies that (1) the numbers of edges are all constant 𝑚, that (2) 𝐹(𝐺_2𝑖) and 𝐹(𝐺_2𝑖+1) are mutually the same finite graphs 𝐺 and 𝐺^′, respectively for all 𝑖, and (3) there are no primitive loops of even length for any graph.
Then, the corresponding sequence of ideals gives an algebraic Markov partition for (𝑉₀,𝑉₁,𝐴) above.

This follows from the following general facts.

Sublemma 6.1
The toric ideal is generated by the set of primitive loops of the even length.

It follows from this that the corresponding ideals are all zero, which defines the affine plane 𝐂^𝑁. This completes the proof.

This simple case suggests that, in general, possibility of construction of algebraic Markov partitions will be reflected by combinatorics of the transcripted configurations.

7.2.2. Automorphism Groups

Let us choose a family of interval maps {𝑓_𝑖}^𝑘, and let 𝑉 be an algebraic variety. We denote 𝐷({𝑓_𝑖}^𝑘,𝑉)≡{({𝑥_𝑗}^𝑙,{𝑎(𝑖,𝑗,ℎ)})∶({𝑓_𝑖}^𝑘,{𝑥_𝑗}^𝑙,{𝑘(𝑖,𝑗,ℎ)})givealgebraicMarkovpartitionsforsome𝐴on𝑉} (7.40) in (∪_𝑙([0,1]^𝑙+1×𝑋₂^𝑘,𝑙).

Let us put the set of the associated automorphisms 𝐸 and the automorphism groups 𝐺 generated by 𝐸 as𝐸({𝑓_𝑖}^𝑘,𝑉)≡{𝐴=𝐴({𝑓_𝑖}^𝑘,{𝑥_𝑗}^𝑘,{𝑘(𝑖,𝑗,ℎ)})∶({𝑥_𝑗}^𝑙,{𝑘(𝑖,𝑗,ℎ)})∈𝐷({𝑥_𝑗}^𝑙,𝑉)},𝐺({𝑓_𝑖}^𝑘,𝑉)=gen𝐸({𝑓_𝑖}^𝑘,𝑉). (7.41) We also put the closure of 𝐺({𝑓_𝑖}^𝑘,𝑉) by 𝐺({𝑓_𝑖}^𝑘,𝑉)⊂ Aut𝑉.

Thus for each algebraic variety 𝑉, one has obtained a map from a set of interval maps and a Lie subgroup of Aut 𝑉: ({𝑓_𝑖}^𝑘,𝑉)→𝐺({𝑓_𝑖}^𝑘,𝑉)⊂Aut𝑉. (7.42)

It is known that, when 𝑉 is compact and nonsingular toric variety, then Aut 𝑉 is linear, and its root system can be written explicitly from the fan of 𝑉. Thus in this case 𝐺⊂ Aut 𝑉 is a closed subgroup of a linear algebraic group. Thus it will be natural to ask the following.

Question 6. Whether one might write down the above subgroups from the information of the associated family of Gröbner fans.

7.3. Zariski Subsets on the Moduli of Interaction Graphs

Let us choose an interaction data: (1) a set of interval maps {𝑓₁,…,𝑓_𝑘}, (2) an index {1,…,𝑙}, and (3) a set of {0,1} sequences {𝑎(𝑖,𝑗,ℎ)}_{𝑖,𝑗,ℎ}⊂𝑋₂.

For each assignment from {1,…,𝑙} to {𝑥₁,…,𝑥_𝑙}, one obtains the associated interaction graph 𝐺. Each edge 𝑒=((𝑖,ℎ),(𝑘,𝑣))∈𝐺 is assigned with some 𝑗∈{1,…,𝑘} so that (𝑓_𝑖,𝑥_ℎ)−−−−−−−−−→^{(𝑓𝑗,𝑎(𝑖,𝑗,ℎ))}(𝑓_𝑘,𝑥_𝑣) hold. So an interaction graph 𝐺_𝑘 is a weighted finite graph such that each edge is assigned with an element in {1,…,𝑘}.

Let us fix 𝑘 and 𝑙 as above and denote the set of interaction graphs as follows: 𝐆(𝑘,𝑙)={𝐺(𝑘,𝑙)∶interactiongraphs}. (7.43) Then 𝐆(𝑘,𝑙) can be parameterized as 𝐆_𝑁≅[0,1]^𝑙×Map[({1,…,𝑘}×{1,…,𝑙})²→{0,1,…,𝑘}]≅[0,1]^𝑀, (7.44) where 0 in the last term implies no edges and 𝑀=𝑀(𝑘,𝑙).

So once one gives an interaction data ({𝑓₁,…,𝑓_𝑘},{𝑎(𝑖,𝑗,ℎ)}), (7.45) then one obtains the family of the associated interaction graphs as follows: 𝐆({𝑓_𝑖}_𝑖,{𝑎(𝑖,𝑗,ℎ)})=_{{𝑥𝑗}𝑗∈[0,1]𝑙}𝐺({𝑓_𝑖}_𝑖,{𝑥_𝑗}_𝑗,{𝑎(𝑖,𝑗,ℎ)}) (7.46) in 𝐆(𝑘,𝑙)≅[0,1]^𝑀.

Definition 7.7. A Zariski subset 𝑋⊂𝐆(𝑘,𝑙)≅[0,1]^𝑀 is a subset of the form 𝑋=𝐆({𝑓_𝑖}_𝑖,{𝑎(𝑖,𝑗,ℎ)})⊂[0,1]^𝑀. (7.47)

Let 𝔄 be all of the set of the interval maps. Then, we have obtained a map 𝐉∶𝔄^𝑘×[0,1]^𝑙×𝑋^𝑘2₂^+𝑙↦𝐆(𝑘,𝑙). (7.48)

Let {𝑓_𝑖} and {𝑔_𝑖} be two 𝑘 interval maps. Then consider the following.

Question 7. (1) In order to guarantee that 𝐉({𝑓_𝑖},)=𝐉({𝑔_𝑖},) implies that {𝑓_𝑖}={𝑔_𝑖}, how should 𝑘 and 𝑙 be large?
(2) Can one find some continuous properties for 𝐉?

Let 𝑋 be a Zariski subset, and consider a proper decreasing Zariski subsets 𝑋=𝑋₀⊃𝑋₁⊃⋯⊃𝑋_𝑛. (7.49) We define the dimension of 𝑋 to be the largest number 𝑛 with the above property.

7.3.1. Dynamics on Zariski Subsets

Let us choose a set {𝑥_𝑗}^𝑙⊂[0,1]^𝑙. Then, one obtains sequences {𝑎(𝑖,𝑗,ℎ)^𝑛}_𝑛=0,1,… by using the interaction map. Thus one obtains a sequence of Zariski subsets as follows:𝑋₀,𝑋₁,…,𝑋_𝑛,𝑋_𝑛+1,…⊂[0,1]^𝑀,𝑋_𝑛=𝐆({𝑓_𝑖},{𝑎(𝑖,𝑗,ℎ)^𝑛}). (7.50) We say that ⋂_𝑛𝑋_𝑛 is an invariant subset. If lim_𝑛𝑋_𝑛⊂𝐆(𝑘,𝑙) exists, then we say that the associated dynamics of the Zariski subsets converges.