Towards a Kneading Theory for Lozi Mappings. II: Monotonicity of the Topological Entropy and Hausdorff Dimension of Attractors

1997 ◽  
Vol 190 (2) ◽  
pp. 375-394 ◽  
Author(s):  
Yutaka Ishii
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Kneading Theory

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.

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.

Isentropic Real Cubic Maps

2003 ◽  
Vol 13 (07) ◽  
pp. 1701-1709 ◽  
Author(s):  
Nuno Martins ◽  
Ricardo Severino ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Level Set ◽  
Topological Invariant ◽  
Symbolic Order ◽  
Kneading Theory ◽  
Theory Framework

Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. With this work, we want to understand how to use this second invariant to distinguish bimodal maps with the same topological entropy and, in particular, how this second invariant changes within a given type of topological entropy level set. In order to do that, we use the kneading theory framework and introduce a symbolic product * between kneading invariants of maps from the same topological entropy level set, for which we show that the second invariant is preserved. Finally, we also show that the change of the second invariant follows closely the symbolic order between bimodal kneading sequences.

DEVIL'S STAIRCASE OF GAP MAPS

2003 ◽  
Vol 13 (01) ◽  
pp. 115-122 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN
Keyword(s):  
Topological Entropy ◽  
Devil's Staircase ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Entropy Functions ◽  
Kneading Theory

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

scholarly journals Symbolic dynamics in mean dimension theory

2020 ◽  
pp. 1-19
Author(s):  
MAO SHINODA ◽  
MASAKI TSUKAMOTO
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Symbolic Dynamics ◽  
Rate Distortion ◽  
Dimension Theory ◽  
Minimal Sets ◽  
Mean Dimension ◽  
The Mean

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$ -subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$ -subshifts with respect to a subaction of $\mathbb{Z}$ . The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$ -subshifts in terms of Kolmogorov–Sinai entropy.

The topological entropy of non-dense orbits and generalized Schmidt games

2017 ◽  
Vol 39 (2) ◽  
pp. 500-530
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Metric Entropy ◽  
Toral Automorphism ◽  
Partially Hyperbolic ◽  
Dense Orbits ◽  
Bounded Below ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

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.

scholarly journals On Hausdorff dimension of invariant sets for expanding maps of a circle

1986 ◽  
Vol 6 (2) ◽  
pp. 295-309 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Cantor Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Cantor Function ◽  
The Family ◽  
Expanding Mapping

AbstractGiven an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

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