scholarly journals The Measure-Theoretic Entropy and Topological Entropy of Actions overℤm

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Chih-Hung Chang ◽  
Yu-Wen Chen
Keyword(s):  
Invariant Measure ◽  
Cellular Automata ◽  
Topological Entropy ◽  
Finite Ring ◽  
One Dimensional ◽  
Maximal Measure ◽  
Bernoulli Measure ◽  
Markov Measures ◽  
Infinite Sequences

This paper studies the quantitative behavior of a class of one-dimensional cellular automata, named weakly permutive cellular automata, acting on the space of all doubly infinite sequences with values in a finite ringℤm,m≥2. We calculate the measure-theoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the spaceℤmℤ. As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.

scholarly journals Conservation Laws and Invariant Measures in Surjective Cellular Automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Jarkko Kari ◽  
Siamak Taati
Keyword(s):  
Cellular Automata ◽  
Conservation Laws ◽  
Arbitrary Dimension ◽  
Close Link ◽  
One Dimensional ◽  
Full Support ◽  
Markov Measure ◽  
Bernoulli Measure ◽  
International Audience ◽  
Markov Measures

International audience We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

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.

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