Bowen Entropy Research Articles

Local fibrations of topological entropy for fibred systems

Given a fibred dynamical system, we introduce the notions of entropy fiber of a fibre for topological entropy, Bowen entropy and packing entropy, which quantifies the “infinitesimal change” in the dynamics of a fibre with respect to its neighboring fibres, this gives rise to an (upper semicontinuous) fibre function. Besides, we show that the topological entropy (Bowen entropy and packing entropy, resp.) of the system is the supremum of the topological entropy fiber (Bowen entropy and packing entropy, resp.) of its fibres, which provides a new perspective on the study of entropy in fibred systems.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

A comparison of topological entropies for nonautonomous dynamical systems

We study the distance entropy, Bowen topological entropy, and the classical topological entropy of nonautonomous dynamical systems. We show, in particular, that the distance entropy, Bowen entropy, Pesin entropy and the classical entropy are equivalent when the system is weakly mixing. Furthermore, we investigate the relationship between distance entropy and Hausdorff dimension on subsets from several aspects in detail.

On Bowen's entropy inequality and almost specification for flows

We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is ergodic. We propose a definition of almost specification property for flows and prove that a continuous flow has the almost specification property if the time-1 map satisfies this property. Using Bowen's inequality for flows, we show that every continuous flow with the almost specification property is saturated, extending a result of Meson and Vericat in [22]. Under the same hypotheses, we extend a result of Thompson on the entropy of irregular points in [29].

Entropy and expansion

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollob\'as's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.

Bowen entropy of sets of generic points for fixed-point free flows

Let (X,ϕ) be a compact metric flow without fixed points. We will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of flows. In this paper, by establishing the Brin–Katok's entropy formula for flows without fixed points in the non-ergodic case, we prove the following result: for an ergodic ϕ-invariant measure μ,htopB(ϕ,Gμ(ϕ))=hμ(ϕ1), where Gμ(ϕ) is the set of generic points for μ and htopB(ϕ,Gμ(ϕ)) is the Bowen entropy on Gμ(ϕ). This extends the classical result of Bowen in 1973 to fixed-point free flows. Moreover, it is shown that the Bowen entropy can be determined via the local entropies of measures.

Measure Expansivity and Specification for Pointwise Dynamics

We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy.

Bowen entropy for fixed-point free flows

In this paper, we devote to the study of the Bowen's entropy for fixed-point free flows and show that the Bowen entropy of the whole compact space is equal to the topological entropy. To obtain this result, we establish the Brin-Katok's local entropy formula for fixed-point free flows in ergodic case.

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Folner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.

The Entropy of Co-Compact Open Covers

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space(compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen's entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces, too.

Entropy and its variational principle for non-compact metric spaces

AbstractIn the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

A Billingsley type theorem for Bowen entropy

For subsets of a metric space with a continuous map, Bowen introduced a notion of entropy. In this Note we show that the Bowen entropy can be determined via the local entropies of measures. This result can be considered as an analogue of Billingsley's Theorem for the Hausdorff dimension. To cite this article: J.-H. Ma, Z.-Y. Wen, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Stable Sets and $$\epsilon$$ -Stable Sets in Positive-Entropy Systems

In this paper, the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated. It is shown that in any positive-entropy system, there is a measure-theoretically “rather big” set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set. Moreover, the Bowen entropy of these weak mixing sets are also estimated. At the same time, it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre-images of $$\epsilon$$ -stable sets, which answers an open question posed by D. Fiebig, U.R. Fiebig and Z.H. Nitecki (Ergod. Th & Dynam. Sys. 23, 1785-1806 (2003)).

Distance entropy of dynamical systems on noncompact-phase spaces

Let $X$ be a separable metric space not necessarily compact, and let $f: X\rightarrow X$ be a continuous transformation. From the viewpoint of Hausdorff dimension, the authors improve Bowen's method to introduce a dynamical quantity distance entropy, written as $ent_{H}(f;Y)$, for $f$ restricted on any given subset $Y$ of $X$; but it is essentially different from Bowen's entropy(1973). This quantity has some basic properties similar to Hausdorff dimension and is beneficial to estimating Hausdorff dimension of the dynamical system. The authors show that if $f$ is a local lipschitzian map with a lipschitzian constant $l$ then $ent_{H}(f;Y)\le\max\{0, $HD$(Y)\log l}$ for all $Y\subset X$; if $f$ is locally expanding with skewness $\lambda$ then $ent_{H}(f;Y)\ge $HD$(Y)\log\lambda$ for any $Y\subset X$. Here HD$(-)$ denotes the Hausdorff dimension. The countable stability of the distance entropy $ent_{H}$ proved in this paper, which generalizes the finite stability of Bowen's $h$-entropy (1971), implies that a continuous pointwise periodic map has the distance entropy zero. In addition, the authors show examples which demonstrate that this entropy describes the real complexity for dynamical systems over noncompact-phase space better than that of various other entropies.

Entropy sets, weakly mixing sets and entropy capacity

Entropy sets are defined both topologically and for a measure. The set of topological entropy sets is the union of the sets of entropy sets for all invariant measures. For a topological system $(X,T)$ and an invariant measure $\mu$ on $(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of the set of all entropy sets (resp. $\mu$- entropy sets) in the hyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp. $(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has an invariant measure with full support and infinite topological entropy. Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropy sets) is a dense $G_\delta$ in $H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed. Concerning entropy capacities, it is shown that when $\mu$ is ergodic with $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ such that the Bowen entropy $h(E)\ge h_\mu(T)$ is residual in $H^\mu(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.

Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen’s definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system. Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.