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$\text{Rot}(\unicode[STIX]{x1D6F7})$.

Entropy and rotation sets: A toy model approach

2016 ◽  
Vol 18 (05) ◽  
pp. 1550083 ◽  
Author(s):  
Tamara Kucherenko ◽  
Christian Wolf
Keyword(s):  
Invariant Measures ◽  
Maximal Entropy ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Lipschitz Continuous ◽  
Exposed Point ◽  
Continuous Potential ◽  
Rotation Set ◽  
Measures Of Maximal Entropy ◽  
Model Approach

Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

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.

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.

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.

scholarly journals Ground states and zero-temperature measures at the boundary of rotation sets

2017 ◽  
Vol 39 (1) ◽  
pp. 201-224
Author(s):  
TAMARA KUCHERENKO ◽  
CHRISTIAN WOLF
Keyword(s):  
Ground State ◽  
Line Segment ◽  
Invariant Measures ◽  
Ground States ◽  
Zero Temperature ◽  
Direction Vector ◽  
Compact Metric Space ◽  
Rotation Vectors ◽  
Continuous Potential ◽  
Rotation Set

We consider a continuous dynamical system $f:X\rightarrow X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states $GS(\unicode[STIX]{x1D6FC})$ of the potential $\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$ as a function of the direction vector $\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\unicode[STIX]{x1D6F7}$. In particular, for each $\unicode[STIX]{x1D6FC}$ the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ forms a non-empty, compact and connected subset of a face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ of the rotation set associated with $\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in \mathbb{N}$ examples with an exposed boundary point (that is, $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ is a non-trivial line segment.

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.

scholarly journals Estimates of topological entropy of continuous maps with applications

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Xiao-Song Yang ◽  
Xiaoming Bai
Keyword(s):  
Topological Entropy ◽  
Simple Proof ◽  
Metric Space ◽  
Simple Theory ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Continuous Maps

We present a simple theory on topological entropy of the continuous maps defined on a compact metric space, and establish some inequalities of topological entropy. As an application of the results of this paper, we give a new simple proof of chaos in the so-calledN-buffer switched flow networks.

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.

Some properties of chain recurrent sets in a nonautonomous discrete dynamical system

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Dhaval Thakkar ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Shadowing Property

AbstractIn this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

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