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 A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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).

THE CANTOR SET’S MULTI-FRACTAL SPECTRUM FORMED BY DIFFERENT PROBABILITY FACTORS IN MATHEMATICAL EXPERIMENT

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750002 ◽  
Author(s):  
XUEZAI PAN ◽  
XUDONG SHANG ◽  
MINGGANG WANG ◽  
ZUO-FEI
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Statistical Physics ◽  
Computer Calculation ◽  
Fractal Dimensions ◽  
The Other ◽  
Mathematical Experiment ◽  
Maximal Width ◽  
Fractal Spectrum ◽  
Vertical Height

With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.

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.

DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1730001 ◽  
Author(s):  
JUN WANG ◽  
KUI YAO
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Box Counting ◽  
Dimension Analysis ◽  
Box Counting Dimension ◽  
Self Similar

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

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).

Sign in / Sign up

Export Citation Format

Share Document