The Set of Lower Semi-Continuity Points of Topological Entropy of a Continuous One-Parametric Family of Dynamical Systems

2019 ◽  
Vol 74 (3) ◽  
pp. 131-133 ◽  
Author(s):  
A. N. Vetokhin
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Parametric Family ◽  
Lower Semi

scholarly journals C0-stability of topological entropy for contactomorphisms

2021 ◽  
pp. 2150015
Author(s):  
Lucas Dahinden
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Topological Entropy ◽  
Small Perturbation ◽  
Geodesic Flows ◽  
Reeb Flows ◽  
Lower Semi ◽  
Special Classes

Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant grows exponentially.

scholarly journals On the Relation between Topological Entropy and Restoration Entropy

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 7 ◽  
Author(s):  
Christoph Kawan
Keyword(s):  
Dynamical Systems ◽  
State Estimation ◽  
Topological Entropy ◽  
Robust Estimation ◽  
Data Transmission ◽  
Communication Constraints ◽  
Dynamical Entropy

In the context of state estimation under communication constraints, several notions of dynamical entropy play a fundamental role, among them: topological entropy and restoration entropy. In this paper, we present a theorem that demonstrates that for most dynamical systems, restoration entropy strictly exceeds topological entropy. This implies that robust estimation policies in general require a higher rate of data transmission than non-robust ones. The proof of our theorem is quite short, but uses sophisticated tools from the theory of smooth dynamical systems.

scholarly journals Topological entropy and partially hyperbolic diffeomorphisms

2008 ◽  
Vol 28 (3) ◽  
pp. 843-862 ◽  
Author(s):  
YONGXIA HUA ◽  
RADU SAGHIN ◽  
ZHIHONG XIA
Keyword(s):  
Topological Entropy ◽  
Topological Invariant ◽  
Compact Manifolds ◽  
Two Dimensional ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Lower Semi ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

scholarly journals Topological Entropy of One Type of Nonoriented Lorenz-Type Maps

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Guo Feng
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Poincaré Map ◽  
Geometric Model ◽  
Dynamical Property ◽  
High Dimensional ◽  
Poincare Map ◽  
One Dimensional

Constructing a Poincaré map is a method that is often used to study high-dimensional dynamical systems. In this paper, a geometric model of nonoriented Lorenz-type attractor is studied using this method, and its dynamical property is described. The topological entropy of one-dimensional nonoriented Lorenz-type maps is also computed in terms of their kneading sequences.

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