Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Spectrum of Dimensions for Escape Time

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

Download Full Text PDF

 

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.

References

  1. [1]  Pesin, Ya. (1997), Dimension Theory in Dinamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, The University of Chicago Press.
  2. [2]  Afraimovich, V., Ugalde, E., and Urías, J. (2006), Fractal Dimensions for Poincaré Recurrences, Monograph Series on Nonlinear Science and Complexity, 2, ELSEVIER.
  3. [3]  Demers, M. and Young, L.S. (2006), Escape rates and conditionally invariant measures, Nonlinearity, 19, 377- 397.
  4. [4]  Homburg, A.J. and Young, T. (2002), Intermittency in families of unimodal maps, Ergodic Theory and Dynamical Systems, 12, 203-225.
  5. [5]  Afraimovich, V. and Bunimovich, L. (2010),Which hole is leaking the most: a topological approach to study open systems, Nonlinearity, 23, 1-14.
  6. [6]  Walters, P. (1976), A variational principle for the pressure of continuous transformations, The American Journal. of Mathematics, 97, 937-971.
  7. [7]  Afraimovich, V. and Hsu, S.B. (2003), Lectures on Chaotic Dynamical Systems, AMS/IP Studies in Advanced Mathematics, 28.
  8. [8]  Moran, P. (1946), Additive functions of intervals and Hausdorff dimension, Mathematical Proceedings of the Cambridge Philosophical Society, 42, 15-23.