New and old facts about entropy in uniform spaces and topological groups (original) (raw)
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Topological entropy can be an indicator of complicated behavior in dynamical systems. It is first introduce by Adler, Konheim and McAndrew by using open covers in 1965. After that it is still an active research by many researchers to produce more properties and applications up to nowadays. The purpose of this paper is to review and explain most important concepts and results of topological entropies of continuous self-maps for dynamical systems on compact and non-compact topological and metric spaces. We give proofs for some of its elementary properties of the topological entropy. Slight modification on Adler's topological entropy is also presented.
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We prove that if a uniformly continuous self-map fff of a uniform space has topological specification property then the map fff has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. An example is also provided to justify that the converse is not true.<br /><br />
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Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen's entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew's entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew's entropy for compact systems.
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Some remarks on topological entropy of a semigroup of continuous maps
A Mathematical Journal Universidad de La …
A Mathematical Journal Vol. 8, No 2,(63-71). August 2006. Some remarks on topological entropy of a semigroup of continuous maps Andrzej Bis 1 Uniwersytet Lódzki, ul. Banacha 22, 90-238 Lodz, Poland andbis@ math. uni. lodz. pl Mariusz Urbanski 2 University of North Texas, ...
Mathematische Nachrichten
In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f‐invariant measures with zero metric entropy is a set (in the weak topology). In particular, this set is generic if the set of f‐periodic measures is dense in the set of f‐invariant measures. This settles a conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equa...
D S ] 1 9 D ec 2 01 2 The entropy of co-compact open covers
2012
Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew’s topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen’s entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen’s entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen’s entropies, and the proof of this result genera...
Topological entropy for set valued maps
Nonlinear Analysis: Theory, Methods & Applications, 2010
Any continuous map T on a compact metric space X induces in a natural way a continuous map T on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map T is zero or infinity. Moreover, the topological entropy of T | C(X) is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.