Abstract
We consider one family of 2-valued transformations on the interval [0, 1] with measure , endowed with a set of weight functions. We construct invariant measure for this multivalued dynamical system with weights and show the interplay between such systems and masked dynamical systems, which leads to image processing.
1. Introduction
Let be a space with finite measure on -field of subsets of , an integer, , and —some measurable transformations. Consider a set of measurable functions (endowment): A collection is called multivalued dynamical system with weights, and the map with fixed pairs —endowed -transformation (see [1]). Regarding this, we can establish a new measure on : One of the important questions of dynamical system theory is finding an invariant measure .
The endowment plays a role of a parameter which controls measure . On the other hand, could be considered as a probability of choosing and applying the transformation (out of ) to a point in stochastic dynamical system. Finally, as we show further, this parameter can uniquely define some single-valued dynamical system connected to .
In this paper we continue (after [2]) studying the following -transformation of the interval (see Figure 1): with a shift as its parameter. Dynamical system is tightly connected to the theory of -decompositions (see [3–6]).
As a motivation for this paper in introduction we examine two points: invariance of measure for this endowed -transformation and masked dynamical system associated with it.
1.1. Invariance of Measure
Let be the Lebesgue measure on and the Borel -field on . Let also be a measure, absolutely continuous with respect to the Lebesgue measure (), with density and .
According to [1], we endow -transformation with a set of weight functions , such that and . Then we can introduce a new measure on :
There are three independent parameters in the abovementioned construction: density function , shift number , and endowment . Whether we search for endowed transformation for a given measure or a measure for a given transformation , there is a certain relation between these parameters, defined by equality .
Further on, we fix three parameters: , , and , and let be such that
Here we cite the following criterion of existence of invariant measure.
Theorem 1 (see [2]). if and only if the following conditions hold true:
where for , , , for , , , for , , and .
There is no restriction on function on the sets and .
Equations (7)-(8) define function on the interval , and (9) defines endowment . We can revise (9) into more compact and constructive formula: where ( is an integer part of ).
To clarify the meaning of the theorem we give two corollaries from it.
Corollary 2 (see [2]). Given measure there exists endowed -transformation preserving measure if and only if , , and satisfy conditions (7)–(9).
Corollary 3 (see [2]). Given endowed -transformation there exists measure which is preserved by transformation if and only if , , and satisfy conditions (7)–(9).
There is a convenient graphical scheme of summation intervals placement on the interval for (7)-(8); see Figures 2 and 3.
Informally, we can depict (7)-(8) as follows:
Regarding Theorem 1, the following question arises.
Question 1. Are there functions satisfying (7)-(8)?
One trivial solution is .
Slightly less trivial example of constant density , , , is presented in the following corollary.
Corollary 4 (see [2]). if and only if , .
However, this -valued dynamical system allows even more sophisticated density: (7)–(9) hold true for some nonconstant , as shown in the next theorem.
Let be a characteristic function for a subset .
Theorem 5 (see [2]). Given there exist a shift (), piecewise constant density , and endowment , such that . Namely,
Remark 6. Theorem 5 yields a family of densities with two parameters .
For computational simplicity in this theorem is chosen in such a way that the middle intervals in the graphical scheme touch each other; see Figure 4 for even .
The resulting piecewise density consists of three domains; see Figure 5.
However, the same question arises again: are there other nontrivial (nonconstant) densities satisfying (7)-(8)?
In Section 2 we present a scheme to construct nontrivial densities in case of , , and study some properties of the functions we obtain there. In Section 3 there is a scheme to construct such densities for arbitrary ().
Finally, in this subsection we cite the following lemma which implies “mirror twoness” of invariant measures densities (see Corollary 8): if is such a density, then the function is again a density of invariant measure.
Lemma 7 (see [2]). Let , . Then if and only if -almost everywhere on
Corollary 8. If is invariant measure density, then the function with endowment , , is also invariant measure density.
Proof. Let be invariant measure density, , and . Substituting instead of in equality (13) yields Thus equality (13) holds true for almost everywhere.
1.2. Masked Dynamical System
As an extra motivation we consider here the following argument: endowment of dynamical system can be connected with mask endowment of some iterated functions system (see below).
Consider some disjoint cover of the set : , , , , , . Let , , be characteristic functions of the subsets .
We may say that, regarding the contribution of to the measure -valued transformation turns into the following single-valued one:
In the case of arbitrary endowment we may consider single-valued stochastic dynamical system:
Such an approach that turns multivalued dynamical system into single-valued one is implemented in [7] for mappings , connected with iterated function systems (IFS). It lets us establish and control fractal transformations between IFS attractors. Such transformations have direct practical value (see below). Here we introduce main points from [7] (relevant to this paper).
Let be a compact Hausdorff space and a set of nonempty compact subsets of . Let be a finite set of positive integers, a set of infinite sequences of numbers from , and , continuous mappings. Then is called iterated function system (IFS).
Due to decreasing monotone inclusion of corresponding compact subsets one can correctly define the mapping If, for all , is a singleton, then the IFS is called point-fibred. In this case a mapping is called the coding map of , the code space of , and the address of the point .
For point-fibred IFS on a compact Hausdorff space there exists a unique set such that and (see [7]). This set is called the attractor of the given IFS.
IFS attractor often happens to be a fractal set or even self-similar one, which is usually of huge interest.
Henceforth, we constrain ourselves to point-fibred IFS on some compact Hausdorff space only (however, this is rather typical, cf. Remark in [7]).
A point may have more than one address (even uncountably many). The following definition will be useful to make the choice of address unique. A subset is called the address space of the IFS if is bijective. Then the inverse mapping is called the section of .
If there are two point-fibred IFS and (with common ) on compact Hausdorff spaces and , and being their attractors, the coding mapping of , and the section of , then we can define the fractal transformation (under this transformation the fractal dimension of a set could be changed) between attractors of and :
The paper [7] gives a continuity criteria for and also describes some applications of fractal transformations for conversion and filtering images and steganography (hidden data transmission, e.g., packing several images into one).
The choice of the address space of defines a fractal transformation. In [7] two methods for construction of are proposed, and they lead to sections with good properties.
One of the methods is to use top addresses: sequences from may be put in lexicographic order, which lets us choose a unique (“top”) element from for all (see [5, 8]). This method is computationally simple and can be easily implemented on computer. However, only a few certain sections can be obtained in this way.
Let us consider the second method in more detail. Let be a point-fibred IFS with injective maps . A collection of subsets is called the mask of if(1);(2); (3).
For all , there exists a unique such that . The mapping is called the masked dynamical system for .
This system is used to construct a section by following the orbit of point ; namely, In this case (see [7]).
Thus the mask of dynamical system connected with IFS is a special case of endowment , when , . We can also consider stochastic mask defined by endowment weight functions: if , , then
Let us describe the connection between this mask construction and -transformation . Consider the following IFS (see Figure 6):
(a)
(b)
This is point-fibred IFS with injective functions , and its attractor is the interval . Consider for construction of masked dynamical system . As might be seen on Figure 6, this dynamical system is the object of this paper. Let be a mask of this IFS. Then obviously, and . Define and arbitrarily (, ). The example of a mask and the process of finding masked address of a point are illustrated on Figure 7.
As we have already mentioned, mask endowment of in this case coincides with endowment of .
Then the following question arises.
Question 2. Is there an invariant measure for this masked dynamical system?
We give an example of such a measure in Section 2.
2. The Case of
Here we consider the case of in detail. The main ideas of this section can be used further for other values of . The conditions (7)-(8) now can be written as or in equivalent way,
To make it simple, we consider special shift, according to the scheme on Figure 4. In our case , ; see Figure 8.
Here we introduce a scheme to construct a density satisfying equations (29)-(30); see Figure 9. Consider the following marks on the -axis: , , , , , , ().(i)Fix functions , , arbitrarily, and define (ii)Fix function , , arbitrarily, and define (iii)Fix function , , arbitrarily, and define (iv)Define for each (v)Fix the value arbitrarily.
By construction, satisfies the conditions (29)-(30) (perhaps except at the most countable number of points on intervals boundaries). Notice that the function is defined arbitrarily on and is restored on after that. We need the partition of the function to study its properties in more detail.
Proposition 9. If are constants, then is a constant, and are constants.
Proof. We denote ; then
Consider the following difference:
To simplify the calculations henceforth, we need the following equalities:
Thus the last expression in equalities (36) equals zero.
Then , .
However, the values of function we obtain can be negative. In the case of piecewise constant density we give the following criterion for to be nonnegative.
Proposition 10. Let be constants, and function is obtained according to the scheme above. Then for all if and only if These inequalities define unbounded convex set in .
Proof. In view of (37), it is sufficient to notice that
Now consider obtaining a function with the property of continuity. This is discussed in Propositions 11–14.
Proposition 11. Given function obtained by the scheme above, then is continuous on if and only if (i)parameters satisfy the equation (ii) is arbitrary;(iii)graphs of continuous functions connect points , , , , and ; see Figure 11.
Proof. Let be continuous; then we substitute into (29)-(30) and obtain
wherefrom . By substituting into (30) and taking into account , we have
which is equal to (40).
To prove the backward implication, let be a piecewise function, made of functions “glued together.”
By construction, on . Then is continuous, because is continuous, and
Now add function leftside into the set of functions which define .
By construction, on , wherefrom is continuous, and (considering )
For , by construction, + on , where is made of functions . Thus is continuous, and
Proposition 12. Let satisfy (29)-(30). If is continuous, then .
Proof. It suffices to show . We substitute into (29) and obtain wherefrom we have (considering (42)).
However, the next question arises.
Question 3. Under which conditions does our construction yield ?
Notice that if by construction of density the equality is not fulfilled, then is continuous on but does not have finite limit at . Its graph is unbounded and (or) oscillates greatly in neighborhood of .
Such situation is quite typical while constructing ; see Figure 12. However, the following Proposition 13 gives an example of a density with good properties.
Proposition 13. Let function be obtained according to the scheme above. If functions form a spline of degree with , then function is also a spline of degree ; furthermore (i) is continuous function;(ii) are linear functions;(iii)graphs of the functions make up one graph of linear function, which connects points and ; see Figure 13.
Proof. Again let , and denote the slope of spline on corresponding intervals by (for ). Then from the construction scheme of itself, we obtain formulae equal to those from the proof of Proposition 9:
Substituting by in (36), we get
We need to show that the last multiplier equals zero. Let , , , , and . We use equalities (37) again:
Taking into account (37), we have
Then , .
Since the second statement of Proposition 11 holds true (by the construction scheme of the spline), function is continuous on . Since it is linear on , then limit exists, and, by Proposition 12, .
Figure 14 shows an example of nontrivial density discussed in Proposition 13. Obviously such function is integrable. To accomplish the topic, we add nonnegativity criterion.
Proposition 14. Let satisfy the conditions of Proposition 13. Denote , , . Then for all if and only if These inequalities define unbounded convex set in .
Proof. Let . In view of Proposition 13, this statement is equivalent to nonnegativeness of spline values at the vertices , .
Let , and substitute into (30). Then using , we have
We use expression (37) to simplify the quantities henceforth:
wherefrom if and only if .
Further, (); thus condition holds true if .
To get the last restriction of the proposition, we consider an equality
Since , then
Since , then , and
Thus
Finally, if and only if , which leads to () and equals
Notice that functions in Figure 12 differ a little from those in Figures 10 and 14: is slightly changed in both cases. Such change leads to great oscillation and (or) unboundedness of . We formulate here the following questions.
(a)
(b)
(a)
(b)
(a)
(b)
Question 4. Explain such “bad” behavior of function . Are there locally nonlinear densities (for them we need to check conditions (29)-(30), nonnegativeness and integrability of )?
Question 5. How can we provide in cases above ( is derived from (9))?
In conclusion, consider the case when is an endowment by characteristic functions of sets of IFS (26) mask. Let be the IFS mask: and , and are arbitrary (, ).
Let , . Condition (9) for turns into On the set , , and (59) implies Similarly, on the set , , and (59) yields
One can see that condition (27) splits into two (as sketched in Figure 15):
Thus we can introduce the following scheme of construction , which is slightly changed version of the one above.
Consider the following marks on the -axis: , , see Figure 16.(i)Fix functions , , arbitrarily, and define (ii)By induction on , define (iii)By induction on , define (iv)Fix values arbitrarily.
Thus function is completely defined by its values on , which are defined by functions and .
Here next question arises.
Question 6. Under which conditions do and ? Is it possible to construct such function for any mask ?
The examples of two masks (see Figure 17) are the partial answer to it. In these examples masks are connected with partition structure of interval over iteration process of density construction. Namely, if , (), , and , then one can show that , outside . If , , and , then , , and outside .
(a)
(b)
We have not found an example of density for arbitrary mask (for instance, that in Figure 7): the function constructed had negative values, unbounded and (or) oscillated greatly.
3. The Case of
In conclusion of the paper, we introduce one of the possible construction schemes for density for all and any :
Consider the following variables (see Figure 18):
Lemma 15. There exists a unique number , , such that , .
Proof. Since , it is sufficient to consider the chain of inequalities:
According to (67), we have ; hence . Then (69) completes the proof.
Now we introduce the following scheme to construct , with (7)-(8) satisfied.
(i) Fix , , arbitrarily, and define
(ii) For , define (see formula (8))
(iii) By induction on , define (see (8))
(iv) By induction on , define (see (7))
(v) Fix value arbitrarily.
Here the following question appears.
Question 7. Under which conditions do and ? Under which conditions does ( is derived from (9))?
Obvious “mirror” change of this scheme is shown in Figure 19 (replacing by ); compare with Corollary 8.
4. Conclusion
Section 1 contains motivation part. It overviews previously derived criteria of measure invariance and some related results, as well as connection between endowment and mask. In Section 2 we consider the case of , and example of mask is given. Section 3 introduces construction scheme for densities with arbitrary .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work was performed according to the Russian Government Program of Competitive Growth of Kazan Federal University as part of the OpenLab “Raduga.” The author expresses his gratitude to K. B. Igudesman for drawing attention to the connection between masks and endowments.