The global statistics of return times: return time dimensions versus generalized measure dimensions

Details The global statistics of return times: return time dimensions versus generalized measure dimensions The global statistics of return times: return time dimensions versus generalized measure dimensions

2009-07-27

arxiv.org/abs/0907.4684v1

J. Stat. Phys. 138 (2010) 701-727

Mathematics

Dynamical Systems

Submitted to J. Stat. Phys

Scientific paper

10.1007/s10955-009-9894-y

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.

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