Pre-image Entropy Research Articles

Parametric topological entropy on orbits of arbitrary multivalued maps in compact Hausdorff spaces

The Adler-Konheim-McAndrew type definitions and the Bowen-Dinaburg-Hood type definitions of parametric topological entropy will be considered on orbits and coincidence orbits of nonautonomous multivalued maps in compact Hausdorff spaces. Their mutual relationship and their link to various further types of definitions like those of (parametric) preimage entropy, will be investigated. In this way, several recent results of the other authors will be generalized and extended into a new setting.

Variational principles for pointwise preimage entropies

Based on the preimage structure of the system , Hurley introduced the notion of pointwise topological preimage entropies and . Furthermore, from the measure-theoretic point of view, Wu and Zhu introduced a notion of pointwise metric preimage entropy for a T-invariant measure µ on X, and obtained the variational principle between and under the condition of uniform separation of preimages. A natural question is whether a variational principle for and without any additional assumptions. In this paper, we define a new version of topological preimage entropy relative to a T-invariant measure µ, and show that the inequality holds for every T-invariant probability measure µ. As a consequence, we obtain that there is a topological dynamical system such that the following strict inequality holds: where denote the set of all T-invariant probability measures.

On entropy and pressure via preimage structure for random dynamical systems

In this paper, entropy, pressure and equilibrium states are investigated for noninvertible random dynamical systems via the preimage structure. For a continuous random dynamical system F on a compact metric space and an F-invariant measure μ, the folding entropy Fμ(F) and the preimage entropy hm,μ(F) are introduced, and it is shown that these two measure-theoretic versions of entropies coincide with each other if F has uniform separation of preimages. Some fundamental properties for hm,μ(F) such as Shannon-McMillan-Breiman Theorem, Kolmogorov-Sinai Theorem and upper semi-continuity are established. From the topological viewpoint, the preimage pressures Pm(F,⋅), Pp(F,⋅) and Pp˜(F,⋅) (resp. topological preimage entropy hm(F), hp(F) and hp˜(F)) are introduced via the preimage structure of single points. Some basic properties are discussed, and particularly, variational principles are obtained. For a C1 nondegenerate random partially hyperbolic dynamical system F on a closed Riemannian manifold, we propose the notions of metric stable entropy hm,μs(F) and stable pressures Pms(F,⋅) and Pps(F,⋅) (resp. topological stable entropies hms(F) and hps(F)) via the preimage structure of local stable manifolds. Several basic properties are obtained and variational principles are established particularly. At the end, as an application of variational principles, p-equilibrium states (resp. s-equilibrium states) and p-tangent functionals (resp. s-tangent functionals) are studied.

Entropy via preimage structure

To measure the complexity of a “non-invertible” continuous map f on a compact metric space via the preimage structure, Hurley introduced the notion of pointwise topological preimage entropies hm(f) and hp(f) in 1995. A natural question was: Can one introduce the counterpart of hm(f) or hp(f) from the measure-theoretic point of view for an f-invariant measure μ and obtain properties analogous to that for the classical entropies? Cheng and Newhouse made the first step on this topic and introduced the notions of preimage entropies hpre(f) and hpre,μ(f) and investigated their properties in 2005. Recently, Wu and Zhu proposed a notion of pointwise metric preimage entropy hm,μ(f) and obtained some properties for f with uniform separation of preimages.In this paper, we give a complete answer to the above question for general continuous maps. The Shannon-McMillan-Breiman theorem for hm,μ(f) and the variational principle relating hm,μ(f) and hm(f) are obtained. It is shown that hm(f)=hpre(f) and the three types of measure-theoretic entropy-like invariants hpre,μ(f), hm,μ(f) and the folding entropy Fμ(f) (introduced by Ruelle) via the preimage structure coincide with each other. We also show that the preimage entropy variational principle can be realized as a variant of the conditional variational principle. These results answer several open questions asked by Cheng-Newhouse and Downarowicz. The upper semi-continuity of hm,μ(f) with respect to invariant measures with the same separation rate is obtained. Moreover, the preimage pressure and preimage equilibrium states are investigated.

On Fractal Properties for Pre-image Entropy

Fractal dimension for pre-image entropy is introduced for continuous maps throughout this paper. First we show the definition of pre-image entropy dimension of a dynamical system from different topological versions. Then we give those basic propositions of pre-image entropy dimension and the formula for power inequality and forward generator. Relationships among different types of pre-image entropy dimension are studied and an inequality relating them is given. Some basic examples are provided to compare those values of polynomial growth type with the pre-image entropy dimension. After that, this study constructs a symbolic subspace to attain any value between 0 and 1 for pre-image entropy dimension.

Preimage Entropy Dimension of Topological Dynamical Systems

We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension satisfies various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy.

Some results on entropy dimension for non-autonomous systems

In this paper, the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced. Moreover, formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given. Finally, examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

On preimage entropy, folding entropy and stable entropy

For non-invertible dynamical systems, we investigate how ‘non-invertible’ a system is and how the ‘non-invertibility’ contributes to the entropy from different viewpoints. For a continuous map on a compact metric space, we propose a notion of pointwise metric preimage entropy for invariant measures. For systems with uniform separation of preimages, we establish a variational principle between this version of pointwise metric preimage entropy and pointwise topological entropies introduced by Hurley [On topological entropy of maps. Ergod. Th. & Dynam. Sys.15 (1995), 557–568], which answers a question considered by Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113]. Under the same condition, the notion coincides with folding entropy introduced by Ruelle [Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys.85(1–2) (1996), 1–23]. For a $C^{1}$-partially hyperbolic (non-invertible and non-degenerate) endomorphism on a closed manifold, we introduce notions of stable topological and metric entropies, and establish a variational principle relating them. For $C^{2}$ systems, the stable metric entropy is expressed in terms of folding entropy (namely, pointwise metric preimage entropy) and negative Lyapunov exponents. Preimage entropy could be regarded as a special type of stable entropy when each stable manifold consists of a single point. Moreover, we also consider the upper semi-continuity for both of pointwise metric preimage entropy and stable entropy and give a version of the Shannon–McMillan–Breiman theorem for them.

Variational principles of partial pre-imageentropy and conditional pre-image entropy

In this paper, wedefine and study two entropy-like invariants for non-invertible map,which are called partial pre-image entropy and conditional pre-imageentropy, respectively, and prove some variational principles forthem. More precisely, let $(X,~T)$ be a topological dynamicalsystem and $\xi$ be a measurable partition of $X$ with $T^{-1}\xi\preceq~\xi$. Then$h_{\mathrm{top}}(T\mid~[\xi]^-)~\geq~\sup_{\mu~\in~\mathcal{M}(X,~T)}h_{\mu}(T\mid~[\xi]^-).$ Moreover, if $[\xi]^-$ can be generated by a upper semi-continuous partition of $X$, then$h_{\mathrm{top}}(T\mid~[\xi]^-)=\sup_{\mu~\in~\mathcal{M}(X,~T)}h_{\mu}(T\mid~[\xi]^-).$ As an application, we study the relativizationof pre-image entropy and obtain other formulas about Cheng-Newhousepre-image entropy.

Pre-image entropy for free semigroup actions

The purpose of this study is to indicate fundamental propositions of the pre-image entropy related to a system proposed by Bufetov [Bufetov A. Topological entropy of free semigroup actions and skew-product transformations. J Dyn Control Syst 1999;5:137–143. 1.] for generating free semigroup actions. This study reveals the formula for the pre-image entropy of skew-product transformation with respect to the one-sided shift space. Finally, one example is presented to show how to obtain the pre-image entropy value for the skew-product transformation.

Preimage pressure for random transformations

AbstractIn this paper, preimage pressure, which is based on the preimage structure of the system, is defined and studied for random transformations. We obtain analogs of many known results of preimage entropy and preimage pressure for deterministic cases in Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113] and Zeng et al [Pre-image pressure and invariant measures. Ergod. Th. & Dynam. Sys.27 (2007), 1037–1052]. In particular, a variational principle is given and some applications of preimage pressure, such as the investigation of the invariant measures and the equilibrium states, are obtained.

On Preimage Entropy of a Sequential Mapping

Biś [2] introduced the notion of entropy-like invariants as an extension of the concept of pointwise preimage entropy. This paper follows Biś concept and extends the behaviour of preimage entropy to the case of a fixed sequence of maps. Therefore, some results are obtained, and we also show that for any continuous maps S and T from a compact metric space onto itself, the maps S ◦ T and T ◦ S have the same pointwise preimage entropy.

PRE-IMAGE ENTROPY OF NONAUTONOMOUS DYNAMICAL SYSTEMS*

The authors define and study topological pre-image entropy for the non-autonomous discrete dynamical systems given by a sequence ${fi}_{\infty}^{i=1}$ of continuous self-maps of a compact topological space. The basic properties and the invariant with respect to equiconjugacy of pre-image entropy for the non-autonomous discrete dynamical systems are obtained.

Preimage entropy for random dynamical systems

In [6], Cheng and Newhouse introduced and studied the newinvariants - preimage entropies for deterministic dynamical systems.In this paper, the analogous notions, measure-theoretic preimageentropy and topological preimage entropy, are formulated for randomdynamical systems. Analogues of many known results for entropies,such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-SinaiTheorem, the Abromov-Rokhlin formula and the power rule, areobtained for preimage entropies. In particular, a variationalprinciple is given.

Two-point pre-image entropy

The aim of this paper is to define and study a new kind of entropy-like invariants in the case of probability space and compact metric topological group of continuous endomorphisms. These new invariants are only non-zero for non-invertible maps, but many propositions can be described and the analogue of the well-known variational principle can be established.

Forward generator for preimage entropy

We investigate preimage entropy and show how to calculate it under the assumption of forward expansiveness. We then define new invariants of non-invertible maps, called the upper preimage entropy and the metric preimage entropy, and obtain variational principles for them. Last, we prove a similar result for the Kolmogorov-Sinai generator.

Pre-image entropy

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

Entropy and preimage sets

We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an -stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h-expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurley's invariants differ and of homeomorphisms with no entropy points.