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Discontinuity, Nonlinearity, and Complexity 2(3) (2013) 247--262 | DOI:10.5890/DNC.2013.08.003

Valentin Afraimovich; Rosendo Vazquez

Instituto de Investigacion en Comunicacion Optica, Universidad Autonoma de San Luis Potosi, Karakorum 1470, Lomas 4a , San Luis Potosi, S.L.P, 78220, Mexico

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Abstract

We introduce a new notion-the spectrum of dimensions of escape time- and study its properties. The escape time was defined for an initial point of the trajectory according to its ability to reach a hole in the phase space. In the article we generalize this notion onto ”spots” of initial points making the escape time to be a function of a set (spot). Then we apply the Caratheodory-Pesin machinary of fractal dimensions to define the spectrum. For dynamical systems generated by maps of the interval or the circle we obtain explicit formulas in the case where an element of Markov partition is chosen as a hole.

Acknowledgments

V. A. was partially supported by PROMEP, UASLP-CA21, R.V. was supported by BECAS CONACyT. The authors like to thank L. Glebsky for useful discussions.

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