Abstract
We compute the limit distribution of the recurrence and of the normalizedk th return times to small sets of the Sierpinski carpet with respect to a natural measure defined on it. It is proved that this dynamical system follows the Poisson law, as one could have expected for such schemes. We study different sequences which converge in finite distribution to the Poisson point process. This limit in law is very interesting in ergodic theory, and it seems to appear for chaotic dynamical systems such as the one we study.
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References
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1973).
M. Hirata, Poisson law for Axiom A diffeomorphisms,Ergod. Theory Dynam. Syst. 13:533–556 (1993).
A. O. Lopez, Entropy and large deviation,Non linearity 3:527–546 (1990).
C. McMullen, The Hausdorff dimension of general Sierpinski carpets,Nagoya Math. J. 96:1–9 (1984).
B. Pitskel, Poisson limit for Markov chains,Ergod. Theory Dynam. Syst. 11:501–513 (1991).
D. Simpelaere, Multifractal decomposition of the Sierpinski carpet, Preprint (1993).
Ya. G. Sinaï, Some mathematical problems in the theory of quantom chaos,Physica A 163:197–204 (1990).
Ya. G. Sinaï, Mathematical problems in the theory of quantom chaos, Preprint.
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Simpelaere, D. Recurrence and return times of the Sierpinski carpet. J Stat Phys 77, 1099–1103 (1994). https://doi.org/10.1007/BF02183155
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DOI: https://doi.org/10.1007/BF02183155