Topological entropy of a dynamical system on the space of one-dimensional maps

2004 ◽  
Vol 7 (2) ◽  
pp. 179-186 ◽  
Author(s):  
S. F. Kolyada
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
One Dimensional ◽  
One Dimensional Maps

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.

TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2001 ◽  
Vol 11 (05) ◽  
pp. 1443-1446 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Topological Entropy ◽  
Positive Entropy ◽  
One Dimensional ◽  
Interval Map ◽  
One Dimensional Maps

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down considerably if the perturbation is small.

Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps

Nonlinearity ◽  
2008 ◽  
Vol 21 (11) ◽  
pp. 2555-2567 ◽  
Author(s):  
Ming-Chia Li ◽  
Ming-Jiea Lyu ◽  
Piotr Zgliczyński
Keyword(s):  
Topological Entropy ◽  
One Dimensional ◽  
One Dimensional Maps

Entropy in dimension one

2014 ◽  
Author(s):  
William P. Thurston
Keyword(s):  
Topological Entropy ◽  
Finite Interval ◽  
Algebraic Integer ◽  
Small Letter ◽  
One Dimensional ◽  
Original Manuscript ◽  
Postcritically Finite ◽  
Perron Number ◽  
One Dimensional Maps ◽  
Dimension One

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.

Periodic points and topological entropy of one dimensional maps

1980 ◽  
pp. 18-34 ◽  
Author(s):  
Louis Block ◽  
John Guckenheimer ◽  
Michal Misiurewicz ◽  
Lai Sang Young
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps

ON THE MONOTONICITY OF ENTROPY FOR MULTILAYER CELLULAR NEURAL NETWORKS

2009 ◽  
Vol 19 (11) ◽  
pp. 3657-3670 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHIH-HUNG CHANG
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Topological Entropy ◽  
Cellular Neural Networks ◽  
Strict Monotonicity ◽  
One Dimensional ◽  
Number Of Layers ◽  
Nested Sequence

This work investigates the monotonicity of topological entropy for one-dimensional multilayer cellular neural networks. The interacting radius and number of layers are treated as parameters. Fix either one of them; the set of topological entropies grows as a strictly nested sequence with respect to one another. Apart from the comparison of the set of topological entropies, maximal and minimal templates are indicators of a dynamical system. Our results demonstrate that maximal and minimal templates of larger interacting radius (respectively number of layers) dominate those of smaller one. To be precise, the strict monotonicity of topological entropy is demonstrated through the comparison of the maximal and minimal templates as the parameters are varied.

scholarly journals Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

2020 ◽  
pp. 1-34
Author(s):  
M. KESSEBÖHMER ◽  
J. D. M. RADEMACHER ◽  
D. ULBRICH
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Travelling Waves ◽  
Excitable Media ◽  
Skew Product ◽  
Invariant Subset ◽  
One Dimensional ◽  
Markov System ◽  
Positive Topological Entropy ◽  
Shift Dynamics

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$ .

BIRKHOFF SPECTRA FOR ONE-DIMENSIONAL MAPS WITH SOME HYPERBOLICITY

2010 ◽  
Vol 10 (01) ◽  
pp. 53-75 ◽  
Author(s):  
YONG MOO CHUNG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Probability Measures ◽  
One Dimensional ◽  
System A ◽  
Topologically Mixing ◽  
One Dimensional Maps ◽  
Dimension One

We study the multifractal analysis for smooth dynamical systems in dimension one. It is given a characterization of the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.

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