Constructing Multi-Branches Complete Chaotic Maps That Preserve Specified Invariant Density

Abstract

A general formulation for multi-branches complete chaotic maps that preserve a specified invariant density is provided and an implication relationship among this class of maps is revealed. Such relationship helps to derive a whole family of complete chaotic maps that preserve not only the same invariant measure but also the same degree of chaos in terms of Lyapunov component.

1. Introduction

Much progress has been achieved in exploring the probabilistic characteristics of nonlinear dynamics from different aspects over the last ten years. While the most general formulations of unimodal (two-branches) complete chaotic maps that preserve general form of invariant densities have been provided in [1, 2], the characterization of such maps for some special forms of invariant densities are analyzed in [2, 3]. Interests to the unimodal chaotic maps with special analytical structural forms are explored in [4–6] and the relevant statistical dynamics are investigated in [7–9]. However, most these studies are limited to complete chaotic maps with two branches only until recently in [10], where the methodologies developed in [1, 2] are generalized to the cases with multi-branches so that most general formulations are derived. The current research intends to continue the same endeavor through deriving more compact and more simplified formulations and to reveal an important relation among them. The results we obtain do not only provide a practical approach to construct multi-branches complete chaotic maps for arbitrary specified analytical expressible invariant densities but also help in understanding the intrinsic characteristics of such maps.

2. Complete Chaotic Maps

Let 𝐽=[0,1] be the unitinterval. Let 𝐹∶𝐽→𝐽 be a nonsingular map. “Nonsigularity” means that 𝑚(𝐹⁻¹(𝑋))=0 whenever 𝑚(𝑋)=0 for a measurable set 𝑋⊂𝐽, where 𝑚 denotes the Lebesgue measure (all mathematical functions in this paper are assumed to be piecewise continuous and differentiable).

Definition 2.1. A nonsingular map 𝐹∶𝐽→𝐽 is said to be complete if(i)there exists a partition 𝐴_𝐹={𝑎_𝑖}^𝑟_𝑖=0, with 0=𝑎₀<𝑎₁<⋯<𝑎_𝑟−1<𝑎_𝑟=1, of the unit interval 𝐽 such that for each integer 𝑖 (𝑖=1,2,…,𝑟), the restriction 𝐹_𝑖 of 𝐹 to the open interval (𝑎_𝑖−1,𝑎_𝑖) is a monotonically continuous function that can be extended to the closed interval 𝐽_𝑖=[𝑎_𝑖−1,𝑎_𝑖];(ii)𝐹_𝑖(𝑎_𝑖−1)∈{0,1},𝐹_𝑖(𝑎_𝑖)∈{0,1}, and 𝐹_𝑖(𝑎_𝑖−1)+𝐹_𝑖(𝑎_𝑖)=1, for 𝑖=1,2,…,𝑟.

We will call 𝐹_𝑖 extended to 𝐽_𝑖 as ith branch of 𝐹, 𝑠_𝑖=̇𝐹_𝑖(𝑎_𝑖)−𝐹_𝑖(𝑎_𝑖−1) as its sign index to indicate the slopping direction of 𝐹 on ith subinterval 𝐽_𝑖, 𝑖=1,…,𝑟, and 𝑆={𝑠_𝑖}^𝑟_𝑖=1 as the sign pattern of 𝐹={𝐹_𝑖}^𝑟_𝑖=1. In particular, we will let 𝐼₊=̇{𝑖∣𝑠_𝑖>0} and 𝐼₋=̇{𝑖∣𝑠_𝑖<0} be the positive and negative index sets that indicate the set of branches with positive and negative slopes, respectively. For instance, the chaotic map defined by

𝐹𝐹(𝑥)=¹(𝑥)=√3𝑥,𝑥∈[0,13𝐹),₂(𝑥)=43−3𝑥²1𝑥∈[3,23𝐹),₃(𝑥)=9𝑥²−452,𝑥∈[3,1], (2.1) is complete with 𝐴_𝐹={0,1/3,2/3,1}, 𝑆={1,−1,1},𝐼₊={1,3}, and 𝐼₋={2}.

Definition 2.2. A complete map 𝐹 is said to be complete chaotic (fully chaotic) if it is(i)ergodic with respect to the Lebesgue measure,(ii)chaotic in the probabilistic sense, that is, an absolutely continuous invariant density 𝜑 is preserved. In other words, for 𝜇(𝑥)=^𝑥₀𝜑(𝑥)𝑑𝑥,𝑥∈𝐽. (2.2)We have 𝜇(𝑋)=𝜇(𝐹⁻¹(𝑋)) (2.3) for any set of 𝑋⊂𝐽 with 𝑚(𝑋)≠0, where 𝑚 denotes the Lebesgue measure.

As illustrated in Figure 1, for a complete chaotic map 𝐹 defined on 𝐽=⋃^𝑟_𝑖=1𝐽_𝑖, for any 𝑥∈𝐽, 𝐹 has exact 𝑟 preimages, denoted as 𝐹_𝑖⁻¹(𝑥), 𝑖=1,2,…,𝑟. Therefore, for any 𝑥∈𝐽₁, 𝐹₁(𝑥) has 𝑟 preimages in each subinterval 𝐽_𝑖, 𝑖≥1, which are linked with 𝑥 through the branching functions 𝜔_𝑖∶𝐽₁→𝐽_𝑖, defined by

Figure 1 

Illustration of branching function.

𝜔_𝑖(𝑥)≐𝐹_𝑖⁻¹(𝐹₁(𝑥)),∀𝑥∈𝐽₁,𝑖=1,2,…,𝑟. (2.4)

Depending on the sign indices of 𝐹₁ and 𝐹_𝑖, the ith branching function 𝜔_𝑖 is a monotonic function that satisfies the following:

(i)if 𝐹^′_𝑖𝐹^′₁>0, then 𝜔^′_𝑖(𝑥)>0 for almost all 𝑥∈𝐽₁ with 𝜔_𝑖(𝑎₀)=𝑎_𝑖−1 and 𝜔_𝑖(𝑎₁)=𝑎_𝑖;(ii)if 𝐹^′_𝑖𝐹^′₁<0, then 𝜔^′_𝑖(𝑥)<0 for almost all 𝑥∈𝐽₁ with 𝜔_𝑖(𝑎₀)=𝑎_𝑖 and 𝜔_𝑖(𝑎₁)=𝑎_𝑖−1.

In particular, we have 𝜔₁=𝑥 and

𝐹₁(𝑥)=𝜇⁻¹(_𝑗∈𝐼+[𝜇(𝜔_𝑗(𝑥))−𝜇(𝑎_𝑗−1)]+_{𝑗∈𝐼−}[𝜇(𝑎_𝑗)−𝜇(𝜔_𝑗(𝑥))]). (2.5)

Instead of working on the branching function 𝜔_𝑖∶𝐽₁→𝐽_𝑖, it is more convenient to manipulate some functions from 𝐽→𝐽 directly. For this purpose, we can define a special class of monotonic increasing one-to-one function on the unitinterval as

Definition 2.3. A 𝐶¹ function 𝑔∶𝐽→𝐽 is said to be a generating function if 𝑔 meets the following requirements:(i)𝑔(0)=0 and 𝑔(1)=1;(ii)𝑔 is continuous; (iii)𝑔^′(𝑥)>0 for almost all 𝑥∈𝐽.

Let 𝒢 be the set of generating functions. Then for a given branching function 𝜔_𝑖∶𝐽₁→𝐽_𝑖,  𝑖=1,2,…,𝑟, there exists a unique 𝑔_𝑖∈𝒢 such that

𝜔_𝑖𝑎(𝑥)=^𝑖−1+(𝑎_𝑖−𝑎_𝑖−1)⋅𝑔_𝑖(𝑥𝑎₁),if𝐹^′_𝑖𝐹^′₁𝑎>0,_𝑖−(𝑎_𝑖−𝑎_𝑖−1)⋅𝑔_𝑖(𝑥𝑎₁),if𝐹^′_𝑖𝐹^′₁<0. (2.6)

In particular, for 𝑔₁(𝑥)=𝑥, we have 𝜔₁(𝑥)=𝑥.

Therefore, for a complete chaotic map 𝐹, there exists a unique set of generating functions 𝐺={𝑔_𝑖}^𝑟_𝑖=1, where 𝑔₁=𝑥 and 𝑔_𝑖∈𝒢, for 𝑖≠1. Utilizing the generating functions (which all share a common domain 𝐽) overcomes the difficulty in specifying branching functions as in (2.5) (which map 𝐽₁ to other subintervals) and hence has advantage in simplifying the general formulation. It has been shown by current author in [10] that the following.

Corollary 2.4. A complete chaotic map 𝐹={𝐹_𝑖}^𝑟_𝑖=1 preserves a normalized absolutely continuous invariant measure 𝜇 if and only if there exists a set of generating functions 𝐺={𝑔_𝑖}^𝑟_𝑖=1, with 𝑔₁(𝑥)=𝑥 and 𝑔_𝑖∈𝒢, for 𝑖=2,3,…,𝑟, so that each and every branch of 𝐹 can be expressed as 𝐹_𝑖𝜇(𝑥)=⁻¹(^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),if𝐹^′₁>0,𝐹^′_𝑖𝜇>0,₋₁(^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝛼_𝑖−𝜇(𝑥)𝛽_𝑖))),if𝐹^′₁>0,𝐹^′_𝑖𝜇<0,₋₁(1−^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝛼_𝑖−𝜇(𝑥)𝛽_𝑖))),if𝐹^′₁<0,𝐹^′_𝑖𝜇>0,₋₁(1−^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),if𝐹^′₁<0,𝐹^′_𝑖<0, (2.7) where 𝛼_𝑖≐𝜇(𝑎_𝑖) and 𝛽_𝑖≐𝛼_𝑖−𝛼_𝑖−1, for 𝑖=1,2,…,𝑟.

Example 2.5. Upward and downward maps.
A complete map is called upward map if 𝑠_𝑖>0 for all 𝑖. We denote it as 𝐹^(𝑢)={𝐹_𝑖^(𝑢)}^𝑟_𝑖=1. Then (2.7) is simplified to
𝐹_𝑖^(𝑢)(𝑥)=𝜇⁻¹(^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),𝑖=1,2,…,𝑟. (2.8)
A complete map is called downward map if 𝑠_𝑖<0 for all 𝑖. We denote it as 𝐹^(𝑑)={𝐹_𝑖^(𝑑)}^𝑟_𝑖=1. Then (2.7) is simplified to
𝐹_𝑖^(𝑑)(𝑥)=𝜇⁻¹(1−^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),𝑖=1,2,…,𝑟. (2.9)

3. Implication Relationship

According to (2.7), the mapping at each section, 𝐹_𝑖, 𝑖=2,3,…,𝑟, depends on the sign index of the first branch only. Therefore, if the sign index of kth (𝑘>1) branch is changed from 𝑠_𝑘 to 𝑠^′_𝑘, with formulation (2.7), only 𝑘th branch 𝐹_𝑘 needs to be changed. This observation reveals an important implication relationship among the set of complete maps that not only preserve an identical invariant measures but also share some analytical similarities.

Theorem 3.1. Let 𝐹={𝐹₁,…,𝐹_𝑘,…,𝐹_𝑟} and 𝐹={𝐹₁,…,𝐹_𝑘,…,𝐹_𝑟} be two complete maps defined on an identical partition 𝐴_𝐹={𝑎_𝑖}^𝑟_𝑖=0 of 𝐽 with the sign patterns differing only in 𝑘th branch (that is, 𝑠_𝑘=−𝑠_𝑘̃). Then 𝐹 and 𝐹 preserve an identical invariant measure 𝜇 if and only if 𝐹_𝑘(𝑥)=𝐹_𝑘(𝜛_𝑘(𝑥)), (3.1) where 𝜛_𝑘(𝑥)=𝜇⁻¹(𝜇(𝑎_𝑘)+𝜇(𝑎_𝑘−1)−𝜇(𝑥)), (3.2) is referred to as 𝑘th branch implication function.

Proof. Without loss of generality, we assume first 𝑘≠1.
The “if” part follows directly from Corollary 2.4, which suggests that, two maps 𝐹 and 𝐹, which have identical functional forms for all branches other than kth one (𝑘>1), can be generated with a same set of generating functions {𝑔_𝑖}^𝑟_𝑖=1. Consider the case in which 𝑠₁=𝑠₁̃>0, 𝑠_𝑘>0 and 𝑠_𝑘̃<0, the different functional forms resulted from
𝐹_𝑘(𝑥)=𝜇⁻¹(^𝑟_𝑖=1(𝜇(𝑎_𝑖)−𝜇(𝑎_𝑖−1))⋅𝑔_𝑖(𝑔_𝑘⁻¹(𝜇(𝑥)−𝜇(𝑎_𝑘−1)𝜇(𝑎_𝑘)−𝜇(𝑎_𝑘−1)𝐹))),_𝑘(𝑥)=𝜇⁻¹(^𝑟_𝑖=1(𝜇(𝑎_𝑖)−𝜇(𝑎_𝑖−1))⋅𝑔_𝑖(𝑔_𝑘⁻¹(𝜇(𝑎_𝑘)−𝜇(𝑥)𝜇(𝑎_𝑘)−𝜇(𝑎_𝑘−1)))). (3.3)
To have (3.1) hold true, the monotonicity of 𝑔_𝑖, 𝑖=1,2,…,𝑟, implies that the following identity must be held:
𝜇(𝜛_𝑘(𝑥))−𝜇(𝑎_𝑘−1)𝜇(𝑎_𝑘)−𝜇(𝑎_𝑘−1)=𝜇(𝑎_𝑘)−𝜇(𝑥)𝜇(𝑎_𝑘)−𝜇(𝑎_𝑘−1), (3.4) which yields (3.2). The cases associated with 𝑠₁=𝑠₁̃<0 can be similarly verified.
The “only if” part can be seen more straightforward from the Frobenius-perron equation (2.5).
If 𝑠_𝑘>0 and 𝑠_𝑘̃<0, we must have
𝐹₁(𝑥)=𝜇⁻¹(_{𝑖∈𝐼+−{𝑘}}[𝜇(𝜔_𝑖(𝑥))−𝜇(𝑎_𝑖−1)]+𝜇(𝜔_𝑘(𝑥))−𝜇(𝑎_𝑘−1)+_{𝑖∈𝐼−}[𝜇(𝑎_𝑖)−𝜇(𝜔_𝑖𝐹(𝑥))]),₁(𝑥)=𝜇⁻¹(_𝑖∈𝐼₊̃[𝜇(𝜔_𝑖(𝑥))−𝜇(𝑎_𝑖−1)]+𝜇(𝑎_𝑘)−𝜇(𝜔_𝑘(𝑥))+_𝑖∈𝐼̃_−−{𝑘}[𝜇(𝑎_𝑖)−𝜇(𝜔_𝑖(𝑥))]). (3.5)
The assumption that 𝐹_𝑖(𝑥)=𝐹_𝑖(𝑥) for all 𝑖≠𝑘 implies that 𝜔_𝑖=𝜔_𝑖 for all 𝑖≠𝑘 and that
𝐼₊−{𝑘}=𝐼₊̃,𝐼₋=𝐼₋̃−{𝑘}, (3.6) which in turn suggest that 𝜇(𝜔_𝑘(𝑥))−𝜇(𝑎_𝑘−1)=𝜇(𝑎_𝑘)−𝜇(𝜔_𝑘(𝑥)), (3.7) or, equivalently, 𝜔_𝑘(𝑥)=𝜇⁻¹(𝜇(𝑎_𝑘)+𝜇(𝑎_𝑘−1)−𝜇(𝜔_𝑘(𝑥))). (3.8)
Therefore, we have
𝜔_𝑘⁻¹(𝑥)=𝜔_𝑘⁻¹(𝜇⁻¹(𝜇(𝑎_𝑘)+𝜇(𝑎_𝑘−1)−𝜇(𝑥))). (3.9)
The facts of 𝐹_𝑘(𝑥)=𝐹₁(𝜔_𝑘⁻¹(𝑥)) and 𝐹_𝑘(𝑥)=𝐹₁(𝜔_𝑘⁻¹(𝑥)), together with (3.9), thus imply (3.2) in order for the implication relationship (3.1) to hold.
The case in which 𝑘=1 can be proved with the same principle.

Remark 3.2. Due to the fact that 𝜛_𝑘∶𝐽_𝑘→𝐽_𝑘 is a monotonically decreasing diagonal-symmetric function in the sense of 𝜛_𝑘(𝑥)=𝜛_𝑘⁻¹(𝑥), (3.10) we also have 𝐹_𝑘(𝑥)=𝐹_𝑘(𝜛_𝑘(𝑥)), (3.11) which suggests that 𝐹_𝑘 and 𝐹_𝑘 are mutually implied with a common implication function 𝜛_𝑘.

Definition 3.3 (implication relation). Two complete chaotic maps, 𝐹={𝐹_𝑖}^𝑟_𝑖=1 and 𝐹={𝐹_𝑖}^𝑟_𝑖=1 defined on an identical partition 𝐴_𝐹={𝑎_𝑖}^𝑟_𝑖=0 of 𝐽, are said to be mutually implied to each other if there exists a subset of indices 𝐼^𝑚≐{𝑖₁,𝑖₂,…,𝑖_𝑚}, with 1≤𝑚≤𝑟 and 𝑖_𝑘∈{1,2,…,𝑟} for 𝑘=1,2,…,𝑚, so that 𝐹_𝑘𝐹(𝑥)=^𝑘(𝜛_𝑘(𝑥)),𝑘∈𝐼^𝑚,𝐹_𝑘(𝑥),𝑘∉𝐼^𝑚, (3.12) where 𝜛_𝑘 is the kth branch implication function defined in (3.2).

Figure 2 demonstrates the mutual-implication family (mutually implication relation was first revealed in ([9]) for two branches complete chaotic maps).

Figure 2 

Mutually implied family.

By repeatedly applying Theorem 3.1, we arrive at the following

Theorem 3.4. Two mutually implied complete chaotic maps preserve an identical invariant measure.

An important characteristic of the implication relationship is that it is invariant with smooth conjugations; that is, if 𝐹_𝑎 and 𝐹_𝑏 are two complete maps mutually implied by a subset of implication functions 𝜛_𝑘, for all 𝑘∈𝐼^𝑚, then for any one-to-one piecewise 𝐶¹ function ℎ, 𝐹_𝑎≐ℎ⁻¹∘𝐹_𝑎∘ℎ, and 𝐹_𝑏≐ℎ⁻¹∘𝐹_𝑏∘ℎ are mutually implied by 𝜛_𝑘, 𝑘∈𝐼^𝑚, where 𝜛_𝑘≐ℎ⁻¹(𝜛_𝑘(ℎ(𝑥))), for all 𝑘∈𝐼^𝑚. An application of mutual implication relationship in constructing an opposite for a two-segmental complete map is offered in [9].

The implication relationship (the set of implication functions) is uniquely determined by the partition 𝐴_𝐹={𝑎_𝑖}^𝑟_𝑖=0 and the invariant measure preserved 𝜇, but is independent of the generating functions 𝐺. For any complete map 𝐹={𝐹_𝑖}^𝑟_𝑖=1 preserving an invariant measure 𝜇, there exists a unique implied family of 𝐹, denoted as Ψ(𝐹), in which each and every map can be derived from the other through replacing some branches with their implied counterparts. On the other hand, for each and every sign pattern 𝑆={𝑠_𝑖}^𝑟_𝑖=1, there associates one and only one map in an implied family. The size of this family is thus always 2^𝑟. Moreover, for any 𝜃={𝜃_𝑖}^𝑟_𝑖=1∈Ψ(𝐹), either 𝜃_𝑖=𝐹_𝑖 or 𝜃_𝑖=𝐹_𝑖∘𝜛_𝑖, for 𝑖=1,2,…,𝑟. And hence, the family Ψ(𝐹) is uniquely specified by 2𝑟 mappings and will be denoted as Ψ(𝐹)=[𝐹^𝑢_𝑖,𝐹^𝑑_𝑖]^𝑟_𝑖=1, or,

𝐹Ψ(𝐹)=^𝑢1,𝐹^𝑢₁∘𝜛₁𝐹⋮⋮_𝑢𝑖,𝐹^𝑢_𝑖∘𝜛_𝑖𝐹⋮⋮_𝑢𝑟,𝐹^𝑢_𝑟∘𝜛_𝑟=𝐹^𝑑₁∘𝜛₁,𝐹^𝑑₁𝐹⋮⋮_𝑑𝑖∘𝜛_𝑖,𝐹^𝑑_𝑖𝐹⋮⋮_𝑑𝑟∘𝜛_𝑟,𝐹^𝑑_𝑟. (3.13)

Due to the uniqueness of implied family, just substituting (3.2) into (2.8) yields the following.

Theorem 3.5. A complete map 𝐹={𝐹_𝑖}^𝑟_𝑖=1 preserves a normalized absolutely continuous invariant measure 𝜇 if and only if there exists a set of generating functions 𝐺={𝑔_𝑖}^𝑟_𝑖=1, with 𝑔₁(𝑥)=𝑥 and 𝑔_𝑖∈𝒢, for 𝑖=2,3,…,𝑟, so that each and every branch of 𝐹 can be expressed as 𝐹_𝑖𝜇(𝑥)=⁻¹(^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),if𝐹^′_𝑖𝜇>0,₋₁(^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝛼_𝑖−𝜇(𝑥)𝛽_𝑖))),if𝐹^′_𝑖<0,𝑖=1,2,…,𝑟, (3.14) where 𝛼_𝑖≐𝜇(𝑎_𝑖) and 𝛽_𝑖≐𝛼_𝑖−𝛼_𝑖−1, for 𝑖=1,2,…,𝑟.

We further show that any member of an implication family shares the same degree of chaos.

Theorem 3.6. All members of an implication family Ψ(𝐹) have an identical Lyapunov exponent.

Proof. Due to the facts that two conjugated complete maps have an identical Lyapunov exponent and that the implication relationship is invariant of topological conjugation, we can proceed with the complete maps that preserve the uniform invariant density (𝜇(𝑥)≡𝑥).
For a complete map 𝐹 defined on a partition 𝐴_𝐹={𝑎_𝑖}^𝑟_𝑖=0 of 𝐽, its Lyapunov exponent, denoted as Λ(𝐹), is given by Λ(𝐹)=∑^𝑟_𝑖=1𝜆_𝑖(𝐹_𝑖), where
𝜆_𝑖(𝐹_𝑖)≐^𝑎𝑖_𝑎𝑖−1ln(|𝐹^′_𝑖(𝑥)|)𝜑(𝑥)𝑑𝑥,∀𝑖∈{1,2,…,𝑟}. (3.15)
It is thus sufficient to prove that the identity 𝜆_𝑖(𝐹_𝑖)=𝜆_𝑖(𝐹_𝑖) holds true for all 𝑖∈{1,2,…,𝑟}, where 𝐹_𝑖=𝐹_𝑖(𝜛_𝑖).
Indeed, since 𝜇(𝑥)≡𝑥 (and hence 𝜑(𝑥)=1), we have 𝜛_𝑖(𝑥)=𝑎_𝑖+𝑎_𝑖−1−𝑥 so that
𝜆_𝑖(𝐹_𝑖)=^𝑎𝑖_𝑎𝑖−1ln(|𝐹^′_𝑖(𝜛_𝑖)𝜛^′_𝑖|)𝑑𝑥=^𝑎𝑖_𝑎𝑖−1ln(|𝐹^′_𝑖(𝑎_𝑖+𝑎_𝑖−1−𝑥)|)𝑑𝑥=𝜆_𝑖(𝐹_𝑖), (3.16) which completes the proof.

For 𝑔_𝑖(𝑥)=𝑥, 𝑖=1,2,…,𝑟, (3.14) simplifies to

𝐹_𝑖𝜇(𝑥)=⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖),if𝐹^′_𝑖𝜇>0,₋₁(𝛼_𝑖−𝜇(𝑥)𝛽_𝑖),if𝐹^′_𝑖<0,𝑖=1,2,…,𝑟, (3.17) which is nothing but a topological conjugate of complete piecewise linear chaotic map defined by

𝑇_𝑖(𝑥)=𝑥−𝛼_𝑖−1𝛼_𝑖−𝛼_𝑖−1,if𝑇^′_𝑖𝛼>0,_𝑖−𝑥𝛼_𝑖−𝛼_𝑖−1,if𝑇^′_𝑖<0,𝑖=1,2,…,𝑟, (3.18) with 𝐴_𝑇={𝛼_𝑖}^𝑟_𝑖=1.

It is well known that 𝑇={𝑇_𝑖}^𝑟_𝑖=1 preserves the Lebesgue measure. This observation reveals the fact that the class of complete chaotic maps that are conjugated to the complete piecewise linear chaotic maps consists of only a subset of all complete chaotic maps that preserve the same absolute continuous invariant measures.

For the special case in which 𝑔_𝑖(𝑥)=𝑥 for all 𝑖, it is immediately observed that

𝜆(𝐹)=−^𝑟_𝑖=1𝛽_𝑖ln𝛽_𝑖. (3.19)

Finally, we just point out that (3.14) is not the unique way to constructing complete chaotic maps with given 𝜇,𝐴_𝐹, and 𝐺. For instance, we may substitute (3.2) into (2.9) alternatively so as to obtain another implied family given by 𝐹={𝐹_𝑖}^𝑟_𝑖=1, where

𝐹_𝑖𝜇(𝑥)=⁻¹(1−^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝛼_𝑖−𝜇(𝑥)𝛽_𝑖))),if𝐹^′_𝑖𝜇>0,₋₁(1−^𝑟_𝑗=1𝛽_𝑗⋅𝑔_𝑗(𝑔_𝑖⁻¹(𝜇(𝑥)−𝛼_𝑖−1𝛽_𝑖))),if𝐹^′_𝑖<0,𝑖=1,2,…,𝑟. (3.20)

𝐹 defined in (3.20) is identical to 𝐹 defined in (3.14) if and only if 𝑔_𝑖(𝑥)=𝑥 for all 𝑖. However, it is easy to see that 𝐹_𝑖 is nothing but to implement formulation (3.14) with a set of new generating functions 𝐺={𝑔_𝑖}^𝑟_𝑖=1 with 𝑔_𝑖(𝑥)=1−𝑔_𝑖(1−𝑥) for all 𝑖. Therefore, we will concentrate on the formulation (3.14) only. Moreover, we can further show that two families Ψ(𝐹) and Ψ(𝐹) have an identical Lyapunov exponent.

4. Numerical Examples

Example 4.1. Consider a 3-to-1 complete chaotic map with 𝜇(𝑥)=2arcsin(√𝑥)/𝜋 (𝜑(𝑥)=1/(𝜋𝑥(1−𝑥))), 𝐴={0,1/4,3/4,1}, and 𝐺={𝑥,𝑥,𝑥}.
With 𝛼₀=0, 𝛼₁=𝜇(1/4)=1/3, 𝛼₂=𝜇(3/4)=2/3, and 𝛼₃=1, we have 𝛽_𝑖=𝛼_𝑖−𝛼_𝑖−1=1/3 for all 𝑖.
For this simple generating functions, we can apply directly formulation (3.17) to get
𝐹₁^(𝑢)(𝑥)=𝐹₂^(𝑑)(𝑥)=𝐹₃^(𝑢)(𝑥)=𝑓(𝑥)≐𝑥(4𝑥−3)²,𝐹₁^(𝑑)(𝑥)=𝐹₂^(𝑢)(𝑥)=𝐹₃^(𝑑)(𝑥)=𝑔(𝑥)≐(1−𝑥)(1−4𝑥)². (4.1)
Moreover, 𝜆(𝐹)=−∑³_𝑖=1𝛽_𝑖ln𝛽_𝑖=ln3.
The three branching functions are given by
𝜛₁(𝑥)=𝜇⁻¹(𝛼₁+𝛼₀1−𝜇(𝑥))=4+121𝑥−2𝜛3𝑥(1−𝑥),₂(𝑥)=𝜇⁻¹(𝛼₂+𝛼₁𝜛−𝜇(𝑥))=1−𝑥,₃(𝑥)=𝜇⁻¹(𝛼₃+𝛼₂1−𝜇(𝑥))=4+121𝑥+23𝑥(1−𝑥). (4.2)
It is interesting to verify that
𝑓(𝜛_𝑖(𝑥))=𝑔(𝑥),𝑔(𝜛₃(𝑥))=𝑓(𝑥) (4.3) for 𝑖=1,2,3.
The above conclusions are illustrated in Figure 3. In particular, Figure 3(d) provides a typical realized frequency from 𝑓(𝑥)=𝑥(4𝑥−3)².

(a) Invariant measure 𝜇 and its inverse

(b) Branching functions

(c) 𝐹 resulted from 𝐺={𝑥,𝑥,𝑥}

(d) 𝜙(𝑥) and frequency of 𝑓(𝑥)

Figure 3 

Illustration of Example 4.1.

Example 4.2. Consider a 3-to-1 complete chaotic map with 𝜇(𝑥)=𝑥²(𝜑(𝑥)=2𝑥) and 𝐴_𝐹={0,1/3,2/3,1}.
With 𝛼₀=0, 𝛼₁=1/9, 𝛼₂=4/9, and 𝛼₃=1, we have 𝛽₁=1/9, 𝛽₂=1/3, and 𝛽₃=5/9, respectively.
The branching functions turn out to be
𝜛₁(𝑥)=𝜇⁻¹(𝛼₁+𝛼₀−𝜇(𝑥))=19−𝑥²,𝜛₂(𝑥)=𝜇⁻¹(𝛼₂+𝛼₁−𝜇(𝑥))=59−𝑥²,𝜛₃(𝑥)=𝜇⁻¹(𝛼₃+𝛼₂−𝜇(𝑥))=139−𝑥². (4.4)
 (i) Consider first the set of generating functions, 𝐺⁽¹⁾≐{𝑥,𝑥,𝑥}, then (3.17) yields
𝐹₁^(𝑢)(𝑥)=3𝑥,𝐹₁^(𝑑)(𝑥)=𝐹₁^(𝑢)(𝜛₁(𝑥))=√1−9𝑥²,𝐹₂^(𝑢)(𝑥)=3𝑥²−13,𝐹₂^(𝑑)(𝑥)=𝐹₂^(𝑢)(𝜛₂1(𝑥))=3√12−27𝑥²,𝐹₃^(𝑢)(𝑥)=95𝑥²−45,𝐹₃^(𝑑)(𝑥)=𝐹₃^(𝑢)(𝜛₃3(𝑥))=5√5−5𝑥². (4.5) It is straightforward to verify that 𝜆(𝐹)≃0.93689=−∑^𝑟_𝑖=1𝛽_𝑖ln𝛽_𝑖.
(ii) Alternatively, if we consider the set of generation functions 𝐺⁽²⁾≐{𝑥,𝑥²,𝑥}, (3.14) yields
𝐹₁^(𝑢)(𝑥)=𝜇⁻¹(𝛽₁⋅𝑔₁(𝑔₁⁻¹(𝜇(𝑥)−𝛼₀𝛽₁))+𝛽₂⋅𝑔₂(𝑔₁⁻¹(𝜇(𝑥)−𝛼₀𝛽₁))+𝛽₃⋅𝑔₃(𝑔₁⁻¹(𝜇(𝑥)−𝛼₀𝛽₁√)))=𝑥6+27𝑥²,𝐹₂^(𝑢)(𝑥)=𝜇⁻¹(𝛽₁⋅𝑔₁(𝑔₂⁻¹(𝜇(𝑥)−𝛼₁𝛽₂))+𝛽₂⋅𝑔₂(𝑔₂⁻¹(𝜇(𝑥)−𝛼₁𝛽₂))+𝛽₃⋅𝑔₃(𝑔₂⁻¹(𝜇(𝑥)−𝛼₁𝛽₂=1)))323(9𝑥²−1)+9𝑥²−1,𝐹₃^(𝑢)(𝑥)=𝜇⁻¹(𝛽₁⋅𝑔₁(𝑔₃⁻¹(𝜇(𝑥)−𝛼₂𝛽₃))+𝛽₂⋅𝑔₂(𝑔₃⁻¹(𝜇(𝑥)−𝛼₂𝛽₃))+𝛽₃⋅𝑔₃(𝑔₃⁻¹(𝜇(𝑥)−𝛼₂𝛽₃=1)))5(9𝑥²−4)(3𝑥²+2). (4.6)
Accordingly, we have
𝐹₁^(𝑑)(𝑥)=𝐹₁^(𝑢)(𝜛₁(𝑥))=(1−3𝑥²)(1−9𝑥²),𝐹₂^(𝑑)(𝑥)=𝐹₂^(𝑢)(𝜛₂1(𝑥))=34−9𝑥²+23(4−9𝑥²),𝐹₃^(𝑑)(𝑥)=𝐹₃^(𝑢)(𝜛₃1(𝑥))=53(1−𝑥²)(19−9𝑥²). (4.7)
Straight calculation reveals that 𝜆(𝐹)=0.89123.
The above conclusions are illustrated in Figure 4.

(a) Invariant measure 𝜇 and its inverse

(b) Branching functions

(c) 𝐹 resulted from 𝐺={𝑥,𝑥,𝑥}

(d) 𝐹 resulted from 𝐺={𝑥,𝑥2,𝑥}

Figure 4 

Illustration of Example 4.2.