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Recurrence rates for loosely Markov dynamical systems

  • Mariusz Urbański (a1)
Abstract
Abstract

The concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] L. Barreira and B. Saussol , ‘Hausdorff dimensions of measures via Poincaré recurrence’, Comm. Math. Phys. 219 (2001), 443463.

[3] R. Benedetti and C. Petronio , Lectures on hyperbolic geometry (Springer, Berlin, 1992).

[4] M. Boshernitzan , ‘Quantitative recurrence results’, Invent. Math. 113 (1993), 617631.

[5] M. Denker and M. Urbański , ‘Ergodic theory of equilibrium states for rational maps’, Nonlinearity 4 (1991), 103134.

[7] P. Hanus and M. Urbański , ‘A new class of positive recurrent functions’, in: Geometry and topology in dynamics, Contemporary Mathematics Series of the AMS 246 (Amer. Math. Soc., Providence, RI, 1999) pp. 123136.

[8] N. Haydn , ‘Convergence of the transfer operator for rational maps’, Ergodic Theory Dynam. Systems 19 (1999), 657669.

[11] Y. Peres , M. Rams , K. Simon and B. Solomyak , ‘Equivalence of positive hausdorif measure and the open set condition for self-conformal sets’, Proc. Amer. Math. Soc. 129 (2001), 26892699.

[13] P. Walters , An introduction to ergodic theory (Springer, 1982).

[16] M. Urbański , ‘Hausdorif measures versus equilibrium states of conformal infinite iterated function systems’, Period. Math. Hungar 37 (1998), 153205.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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