ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 363-370 ◽  
Author(s):  
DONGKUI MA ◽  
MIN WU
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Topological Space ◽  
Topological Entropy ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Metric Function

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1437-1438 ◽  
Author(s):  
SERGIĬ KOLYADA ◽  
LUBOMÍR SNOHA
Keyword(s):  
Dynamical System ◽  
Topological Space ◽  
Topological Entropy ◽  
Discrete Dynamical System ◽  
Topological Dynamics ◽  
Limit Sets ◽  
Compact Topological Space ◽  
Continuous Maps ◽  
Uniformly Convergent

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence [Formula: see text] of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence [Formula: see text] either forms an equicontinuous family of maps or is uniformly convergent. We also show that for any continuous maps f and g from a compact topological space into itself the topological entropies h(f ◦ g) and h(g ◦ f) are equal.

scholarly journals Entropy and its variational principle for locally compact metrizable systems

2016 ◽  
Vol 38 (2) ◽  
pp. 540-565 ◽  
Author(s):  
ANDRÉ CALDAS ◽  
MAURO PATRÃO
Keyword(s):  
Variational Principle ◽  
Lie Groups ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Invariant Probability Measure ◽  
Compact Set ◽  
Locally Compact ◽  
Finite Dimensional Vector Space ◽  
Continuous Map ◽  
Image Position

For a given topological dynamical system $T:X\rightarrow X$ over a compact set $X$ with a metric $d$, the variational principle states that $$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=h_{d}(T),\end{eqnarray}$$ where $h_{\unicode[STIX]{x1D707}}(T)$ is the Kolmogorov–Sinai entropy with the supremum taken over every $T$-invariant probability measure, $h_{d}(T)$ is the Bowen entropy and $h(T)$ is the topological entropy as defined by Adler, Konheim and McAndrew. In Patrão [Entropy and its variational principle for non-compact metric spaces. Ergod. Th. & Dynam. Sys. 30 (2010), 1529–1542], the concept of topological entropy was adapted for the case where $T$ is a proper map and $X$ is locally compact separable and metrizable, and the variational principle was extended to $$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=\min _{d}h_{d}(T),\end{eqnarray}$$ where the minimum is taken over every distance compatible with the topology of $X$. In the present work, we drop the properness assumption and extend the above result for any continuous map $T$. We also apply our results to extend some previous formulae for the topological entropy of continuous endomorphisms of connected Lie groups that were proved in Caldas and Patrão [Dynamics of endomorphisms of Lie groups. Discrete Contin. Dyn. Syst. 33 (2013). 1351–1363]. In particular, we prove that any linear transformation $T:V\rightarrow V$ over a finite-dimensional vector space $V$ has null topological entropy.

Minimal invariant spaces in formal topology

1997 ◽  
Vol 62 (3) ◽  
pp. 689-698 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Direct Consequence ◽  
Arithmetical Progression ◽  
Standard Result ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Combinatorial Theorem ◽  
Minimal Property ◽  
Special Instance ◽  
Invariant Spaces ◽  
Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

A variation formula for the topological entropy of convex-cocompact manifolds

2010 ◽  
Vol 31 (6) ◽  
pp. 1849-1864 ◽  
Author(s):  
SAMUEL TAPIE
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Formula ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Kleinian Group ◽  
Limit Set ◽  
Analytic Family ◽  
Variation Formula ◽  
Sectional Curvatures ◽  
Convex Cocompact

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

scholarly journals Hausdorff Dimension and Topological Entropies of a Solenoid

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 506
Author(s):  
Andrzej Biś ◽  
Agnieszka Namiecińska
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Fractal Dimensions

The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz solenoids and locally λ — expanding solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally λ — expanding solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of λ .

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

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