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.

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.

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.

Accessibility on iterated function systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Maliheh Mohtashamipour ◽  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Sufficient Condition ◽  
Skew Product ◽  
Iterated Function ◽  
Topologically Mixing ◽  
Mixing Property

AbstractIn this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

Some properties on topological entropy of free semigroup action

2017 ◽  
Vol 33 (1) ◽  
pp. 54-71 ◽  
Author(s):  
Jingru Tang ◽  
Bing Li ◽  
Wen-Chiao Cheng
Keyword(s):  
Topological Entropy ◽  
Free Semigroup ◽  
Semigroup Action

scholarly journals Some Remarks on Measure-theoretic Entropy for a Free Semigroup Action

2017 ◽  
Vol 21 (2) ◽  
pp. 429-440
Author(s):  
Huihui Hui ◽  
Dongkui Ma
Keyword(s):  
Free Semigroup ◽  
Semigroup Action

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.

scholarly journals A bridge theorem for the entropy of semigroup actions

2020 ◽  
Vol 8 (1) ◽  
pp. 46-57
Author(s):  
Anna Giordano Bruno
Keyword(s):  
Abelian Group ◽  
Topological Entropy ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Finitely Generated ◽  
Semigroup Action ◽  
Locally Compact ◽  
Semigroup Actions ◽  
Dual Action ◽  
Totally Disconnected

AbstractThe topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

scholarly journals Quantitative recurrence for free semigroup actions

Nonlinearity ◽  
2018 ◽  
Vol 31 (3) ◽  
pp. 864-886 ◽  
Author(s):  
Maria Carvalho ◽  
Fagner B Rodrigues ◽  
Paulo Varandas
Keyword(s):  
Free Semigroup ◽  
Semigroup Actions ◽  
Quantitative Recurrence
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