Abstract
We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known Pólya–Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.
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Acknowledgements
SV thanks the Laboratoire International Associé LIA LYSM, the INdAM (Italy), the UMI-CNRS 3483, Laboratoire Fibonacci (Pisa) where this work has been completed under a CNRS delegation and the Centro de Giorgi in Pisa for various supports. NH thanks the University of Toulon and the Simons Foundation (Award ID 526571) for support. The authors thanks the anonimous referees whose comments and suggestion helped them to improve the paper.
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Haydn, N., Vaienti, S. Limiting Entry and Return Times Distribution for Arbitrary Null Sets. Commun. Math. Phys. 378, 149–184 (2020). https://doi.org/10.1007/s00220-020-03795-0
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DOI: https://doi.org/10.1007/s00220-020-03795-0