In the present paper, we introduce a natural extension of AKM-topological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the classical formula for the entropy of an automorphism of a noncompact Lie group is just an upper bound for its topological entropy.
We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass phenomenon and the measure theoretic entropy on finite volume nonpositively curved manifolds satisfying the Visibility axiom. We provide a general criterion for the same relation to hold between the escape of mass and the measure theoretic entropy. This gives a criterion for the existence of measures of maximal entropy for the geodesic flow on some nonpositively c…
For a geodesic flow on a negatively curved Riemannian manifold $M$ and some subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set of points whose $ω$-limit do not intersect $A$. We show that if the topological $\ast$-entropy of $A$ is smaller than the topological entropy of the geodesic flow, then the limit $A$-exceptional set has full topological entropy. Some consequences are stated for limit exceptional sets of invariant compact subsets and proper submanifolds.
In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.
In this work, we introduce the notion of entropy at infinity, and define a wide class of noncompact manifolds with negative curvature, those which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy. Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) …
We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowe…
In this note, we show several variational principles for metric mean dimension. First we prove a variational principles in terms of Shapira's entropy related to finite open covers. Second we establish a variational principle in terms of Katok's entropy. Finally using these two variational principles we develop a variational principle in terms of Brin-Katok local entropy.
Ovadia and Rodriguez-Hertz (Neutralized local entropy, arXiv:2302.10874) defined the neutralized Bowen open ball for an autonomous dynamical system on a compact metric space. Replacing the usual Bowen open ball with neutralized Bowen open ball, we introduce the notions of neutralized Bowen topological entropy of subsets, neutralized weighted Bowen topological entropy of subsets, lower neutralized Brin-Katok's local entropy of Borel probability measures, and neutralized Katok's entropy of Borel probability measures for a non-autonomous dynamical system on a compact metric space. Then, we establ…
This paper discusses the variational principles on subsets for topological pressure and topological entropy of non-autonomous dynamical systems. We define the Pesin-Pitskel topological pressure (weighted topological pressure) and the Bowen topological entropy (weighted Bowen topological entropy) for any subset. Also, we define the measure-theoretic pressure and the measure-theoretic lower entropy for all Borel probability measures. Then, we prove variational principles for topological pressure (topological entropy) which links the Pesin-Pitskel topological pressure (weighted topological pressu…
In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $π: (X,d, T)\to (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira's entropy, Katok's entropy and Brin-Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi \cite{Shi} partially for a very well-partitionable compact metric space, and in general w…
In this paper we derived a variational principle for the specific entropy on the context of symbolic dynamics of compact metric space alphabets and use this result to obtain the uniqueness of the equilibrium states associated to a Walters potential.
Variational principle for random pressure function
College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Key Laboratory of Nonlinear Analysis and its Applications (Chongqing University), Ministry of Education（教育部非线性分析及其应用重点实验室（重庆大学））
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 er…
For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimension central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variatio…
In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the $u$-equilibrium states are given.
Let $\mathcal{F}$ be a $C^2$ random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of $\mathcal{F}$ on the unstable foliation are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs $u$-states …
In this paper, we determine the topological entropy $h(ϕ)$ of a continuous endomorphism $ϕ$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(ϕ)$ associated to a left invariant distance $d$ just provides an upper bound to $h(ϕ)$, which is characterized by the so called variational principle. We prove that \[ h\left(ϕ\right) = h\left(ϕ|_{T(G_ϕ)}\right) \] where $G_ϕ$ is the maximal connected subgroup of $G$ such that $ϕ(G_ϕ) = G_ϕ$, and $T(G_ϕ)$ is …
The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle. More precisely, using methods from Convex Analysis, we construct for each generalized convex pressure function an upper semi-continuous entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy),…
Firstly, we answer the problem 1 asked by Gutman and $\rm \acute{\ S}$piewak in \cite{gs20}, then we establish a double variational principle for mean dimension in terms of R$\bar{e}$nyi information dimension and show the order of $\sup$ and $\limsup$ (or $\liminf$) of the variational principle for the metric mean dimension in terms of R$\bar{e}$nyi information dimension obtained by Gutman and $\rm \acute{\ S}$piewak can be changed under the marker property. Finally, we attempt to introduce the notion of maximal metric mean dimension measure, which is an analogue of the concept called classica…
We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (ie, slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (ie, slow entropy) coincides with the topological one, establishing hence variational principle for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system t…
