scholarly journals Transitive dendrite map with zero entropy

2016 ◽  
Vol 37 (7) ◽  
pp. 2077-2083 ◽  
Author(s):  
JAKUB BYSZEWSKI ◽  
FRYDERYK FALNIOWSKI ◽  
DOMINIK KWIETNIAK
Keyword(s):  
Topological Entropy ◽  
Chaotic Maps ◽  
Dendrite Map ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Entropy Estimates

Hoehn and Mouron [Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys.34 (2014), 1897–1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn–Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Entropy estimates for transitive maps on trees. Topology40(3) (2001), 551–569].

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Hierarchies of chaotic maps on continua

2013 ◽  
Vol 34 (6) ◽  
pp. 1897-1913 ◽  
Author(s):  
LOGAN HOEHN ◽  
CHRISTOPHER MOURON
Keyword(s):  
Chaotic Maps ◽  
Weakly Mixing ◽  
Will Force

AbstractLet $f: X\longrightarrow X$ be a map of a continuum. In this paper we examine the following dynamical conditions on $f$: (1) $f$ is continuum-wise fully expansive; (2) $f$ is weakly continuum-wise fully expansive; (3) $f$ is mixing; (4) $f$ is weakly mixing. We first show that (1) implies (2), (2) implies (3) and (3) implies (4). Then we investigate what topological conditions will force the reverse implications to hold and give examples of when the reverse conditions do not hold. In particular, a map of the universal dendrite is given that is weakly mixing but not mixing.

Zero-entropy invariant measures for skew product diffeomorphisms

2009 ◽  
Vol 30 (3) ◽  
pp. 923-930 ◽  
Author(s):  
PENG SUN
Keyword(s):  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Affirmative Answer ◽  
Skew Product ◽  
Hyperbolic Structure ◽  
Combinatorial Properties ◽  
Positive Topological Entropy ◽  
Zero Entropy ◽  
Uniquely Ergodic

AbstractIn this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

scholarly journals Smooth chaotic maps with zero topological entropy

1988 ◽  
Vol 8 (3) ◽  
pp. 421-424 ◽  
Author(s):  
M. Misiurewicz ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Chaotic Maps

AbstractWe find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.

scholarly journals Topological entropy of Devaney chaotic maps

2003 ◽  
Vol 133 (3) ◽  
pp. 225-239 ◽  
Author(s):  
Francisco Balibrea ◽  
L'ubomír Snoha
Keyword(s):  
Topological Entropy ◽  
Chaotic Maps

scholarly journals Loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli

2021 ◽  
pp. 1-57
Author(s):  
MARLIES GERBER ◽  
PHILIPP KUNDE
Keyword(s):  
Structure Theorem ◽  
Classification Problem ◽  
Global Structure ◽  
Classification Result ◽  
Rank One ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Circular System ◽  
Factor Maps

Abstract Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint, 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) mapping odometer-based systems, $\mathcal {OB}$ , to circular systems, $\mathcal {CB}$ . This functor transfers the classification problem from $\mathcal {OB}$ to $\mathcal {CB}$ , and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal {F}$ preserves other dynamical properties. We show that $\mathcal {F}$ does not preserve the loosely Bernoulli property by providing positive and zero-entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

scholarly journals Symbolic dynamics on amenable groups: the entropy of generic shifts

2016 ◽  
Vol 37 (4) ◽  
pp. 1187-1210 ◽  
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Symbolic Dynamics ◽  
Amenable Group ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Strongly Irreducible ◽  
Isolated Points ◽  
Zero Entropy ◽  
Weakly Mixing

Let$G$be a finitely generated amenable group. We study the space of shifts on$G$over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy$c$are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value$c\in [0,\log |A|]$, there is a weakly mixing subshift of$A^{G}$with entropy $c$. We also show that the set of strongly irreducible shifts does not form a$G_{\unicode[STIX]{x1D6FF}}$in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

scholarly journals Positive topological entropy and Δ-weakly mixing sets

2017 ◽  
Vol 306 ◽  
pp. 653-683 ◽  
Author(s):  
Wen Huang ◽  
Jian Li ◽  
Xiangdong Ye ◽  
Xiaoyao Zhou
Keyword(s):  
Topological Entropy ◽  
Positive Topological Entropy ◽  
Weakly Mixing

ZERO ENTROPY PERMUTATIONS

1995 ◽  
Vol 05 (05) ◽  
pp. 1331-1337
Author(s):  
LOUIS BLOCK ◽  
ALEXANDER M. BLOKH ◽  
ETHAN M. COVEN
Keyword(s):  
Topological Entropy ◽  
Graph Theoretic ◽  
Zero Entropy

The entropy of a permutation is the (topological) entropy of the "connect-the-dots" map determined by it. We give matrix- and graph-theoretic, geometric, and dynamical characterizations of zero entropy permutations, as well as a procedure for constructing all of them. We also include some information about the number of zero entropy permutations.

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