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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Entropy formulas for dynamical systems with mistakes

Pages: 4391 - 4407, Volume 32, Issue 12, December 2012      doi:10.3934/dcds.2012.32.4391

 
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Jérôme Rousseau - Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil (email)
Paulo Varandas - Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil (email)
Yun Zhao - Department of Mathematics, Soochow University, Suzhou 215006, China (email)

Abstract: We study the recurrence to mistake dynamical balls, that is, dynamical balls that admit some errors and whose proportion of errors decrease tends to zero with the length of the dynamical ball. We prove, under mild assumptions, that the measure-theoretic entropy coincides with the exponential growth rate of return times to mistake dynamical balls and that minimal return times to mistake dynamical balls grow linearly with respect to its length. Moreover we obtain averaged recurrence formula for subshifts of finite type and suspension semiflows. Applications include $\beta$-transformations, Axiom A flows and suspension semiflows of maps with a mild specification property. In particular we extend some results from [6, 10, 19] for mistake dynamical balls.

Keywords:  Poincare recurrences, entropy, mistake function, topological pressure, suspension semiflow.
Mathematics Subject Classification:  Primary: 37B20; Secondary: 37A35.

Received: June 2011;      Revised: February 2012;      Available Online: August 2012.

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