scholarly journals Uniformly positive entropy of induced transformations

2020 ◽  
pp. 1-10
Author(s):  
NILSON C. BERNARDES ◽  
UDAYAN B. DARJI ◽  
RÔMULO M. VERMERSCH
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Weak Topology ◽  
Probability Measures ◽  
Entropy Theory ◽  
Positive Entropy ◽  
Local Entropy ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
Borel Probability

Abstract Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

On the set of expansive measures

2018 ◽  
Vol 20 (07) ◽  
pp. 1750086 ◽  
Author(s):  
Keonhee Lee ◽  
C. A. Morales ◽  
Bomi Shin
Keyword(s):  
Metric Space ◽  
Weak Topology ◽  
Hausdorff Metric ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Whole Space ◽  
Isolated Points ◽  
Borel Probability ◽  
Heteroclinic Points

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

scholarly journals On the computability of rotation sets and their entropies

2018 ◽  
Vol 40 (2) ◽  
pp. 367-401 ◽  
Author(s):  
MICHAEL A. BURR ◽  
MARTIN SCHMOLL ◽  
CHRISTIAN WOLF
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
Topological Entropy ◽  
Metric Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Subshift Of Finite Type ◽  
Continuous Potential ◽  
Rotation Set ◽  
Entropy Functions

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

Chaos of Transformations Induced Onto the Space of Probability Measures

2016 ◽  
Vol 26 (13) ◽  
pp. 1650227 ◽  
Author(s):  
Xinxing Wu
Keyword(s):  
Dynamical System ◽  
Weak Topology ◽  
Probability Measures ◽  
Weakly Mixing ◽  
Borel Probability ◽  
Uniformly Rigid

For a dynamical system [Formula: see text], let [Formula: see text] be its induced dynamical system on the space of Borel probability measures with weak*-topology. It is proved that [Formula: see text] is [Formula: see text]-transitive (resp., exact, uniformly rigid) if and only if [Formula: see text] is weakly mixing and [Formula: see text]-transitive (resp., exact, uniformly rigid), where [Formula: see text] is an [Formula: see text]-vector of integers. Moreover, some analogous results are obtained for the hyperspace.

scholarly journals Packing topological entropy for amenable group actions

2021 ◽  
pp. 1-35
Author(s):  
DOU DOU ◽  
DONGMEI ZHENG ◽  
XIAOMIN ZHOU
Keyword(s):  
Topological Entropy ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Borel Subset ◽  
Entropy Inequality ◽  
Probability Measures ◽  
Local Entropy ◽  
Compact Metric Space ◽  
Borel Probability ◽  
Generic Points

Abstract Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

scholarly journals Thickly Syndetical Sensitivity of Topological Dynamical System

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Heng Liu ◽  
Li Liao ◽  
Lidong Wang
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Topological Dynamical System ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Devaney Chaos ◽  
Weakly Mixing

Consider the surjective continuous mapf:X→X, whereXis a compact metric space. In this paper we give several stronger versions of sensitivity, such as thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, and strong sensitivity. We establish the following. (1) If(X,f)is minimal and sensitive, then(X,f)is syndetically sensitive. (2) Weak mixing implies thick sensitivity. (3) If(X,f)is minimal and weakly mixing, then it is thickly syndetically sensitive. (4) If(X,f)is a nonminimalM-system, then it is thickly syndetically sensitive. Devaney chaos implies thickly periodic sensitivity. (5) We give a syndetically sensitive system which is not thickly sensitive. (6) We give thickly syndetically sensitive examples but not cofinitely sensitive ones.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

scholarly journals Geometrical properties of the space of idempotent probability measures

2021 ◽  
Vol 22 (2) ◽  
pp. 399
Author(s):  
Kholsaid Fayzullayevich Kholturayev
Keyword(s):  
Metric Space ◽  
Category Theory ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Geometrical Properties ◽  
Idempotent Mathematics

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

Sign in / Sign up

Export Citation Format

Share Document