On the measure-theoretic entropy and topological pressure of free semigroup actions

2016 ◽  
Vol 38 (2) ◽  
pp. 686-716 ◽  
Author(s):  
XIAOGANG LIN ◽  
DONGKUI MA ◽  
YUPAN WANG
Keyword(s):  
Variational Principle ◽  
Free Semigroup ◽  
Topological Pressure ◽  
Skew Product ◽  
Affine Transformations ◽  
Semigroup Action ◽  
Compact Metric Space ◽  
Metrizable Group ◽  
Initial Action ◽  
The Relationship

In this paper we introduce the notions of topological pressure and measure-theoretic entropy for a free semigroup action. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action we assign a skew-product transformation whose fiber topological pressure is taken to be the topological pressure of the initial action. Some properties of these two notions are given, followed by two main results. One is the relationship between the topological pressure of the skew-product transformation and the topological pressure of the free semigroup action, the other is the partial variational principle about the topological pressure. Moreover, we apply this partial variational principle to study the measure-theoretic entropy and the topological entropy of finite affine transformations on a metrizable group.

The multi-transitivity of free semigroup actions

2020 ◽  
Vol 20 (05) ◽  
pp. 2050040
Author(s):  
Zhumin Ding ◽  
Jiandong Yin ◽  
Xiaofang Luo
Keyword(s):  
Free Semigroup ◽  
Skew Product ◽  
Product System ◽  
Semigroup Action ◽  
Semigroup Actions ◽  
Equivalent Conditions ◽  
Mixing Property

In this paper, we introduce the conceptions of multi-transitivity, [Formula: see text]-transitivity and [Formula: see text]-mixing property for free semigroup actions and give some equivalent conditions for a free semigroup action to be multi-transitive, multi-transitive with respect to vectors and strongly multi-transitive, respectively. For instance, we prove that a free semigroup action is multi-transitive or multi-transitive with respect to a vector if and only if its corresponding skew product system is multi-transitive or multi-transitive with respect to the same vector.

scholarly journals A variational principle for the metric mean dimension of free semigroup actions

2021 ◽  
pp. 1-21
Author(s):  
MARIA CARVALHO ◽  
FAGNER B. RODRIGUES ◽  
PAULO VARANDAS
Keyword(s):  
Random Walk ◽  
Variational Principle ◽  
Metric Space ◽  
Free Semigroup ◽  
Borel Probability Measure ◽  
Box Dimension ◽  
Compact Metric Space ◽  
Semigroup Actions ◽  
Mean Dimension ◽  
The Impact

Abstract We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.

Some dynamical properties for free semigroup actions

2018 ◽  
Vol 18 (04) ◽  
pp. 1850032 ◽  
Author(s):  
Huihui Hui ◽  
Dongkui Ma
Keyword(s):  
Metric Space ◽  
Free Semigroup ◽  
Semigroup Action ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Semigroup Actions ◽  
Specification Property ◽  
Chain Transitivity ◽  
Weakly Mixing ◽  
Dynamical Properties

In this paper, we introduce the notions of weakly mixing and totally transitivity for a free semigroup action. Let [Formula: see text] be a free semigroup acting on a compact metric space generated by continuous open self-maps. Assuming shadowing for [Formula: see text] we relate the average shadowing property of [Formula: see text] to totally transitivity and its variants. Also, we study some properties such as mixing, shadowing and average shadowing properties, transitivity, chain transitivity, chain mixing and specification property for a free semigroup action.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

Tail pressure and the tail entropy function

2011 ◽  
Vol 32 (4) ◽  
pp. 1400-1417 ◽  
Author(s):  
YUAN LI ◽  
ERCAI CHEN ◽  
WEN-CHIAO CHENG
Keyword(s):  
Variational Principle ◽  
Invariant Measures ◽  
Direct Proof ◽  
Equilibrium States ◽  
Entropy Function ◽  
Topological Pressure

AbstractBurguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.

scholarly journals Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 309
Author(s):  
Xianfeng Ma ◽  
Zhongyue Wang ◽  
Hailin Tan
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Image Measure ◽  
Measure Preserving Transformations

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.

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