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Recurrence and Lyapunov exponents. (English) Zbl 1083.37504
Let \(f\) be a Borel measurable map of the metric space \((M,d)\) preserving an ergodic invariant probability measure \(\mu\). Define the first return of any set \(A\subset M\) by \[ \tau(A):=\min\{k>0\mid f^k(A)\cap A\neq \emptyset\}.\tag{1} \] The authors generalize formula (2), that is \[ \varliminf_{r\to 0}\frac{\tau(B(x,r))}{-\log r}\geq \frac 1{\lambda\mu}\text{ for }\mu\text{-almost every }x,\tag{2} \] for multidimensional transformations, providing examples which show that the inequalities obtained by the authors are sharp and other examples where strict inequalities hold. By \(\lambda_\mu\) is denoted the positive Lyapunov exponent of the measure \(\mu\) of nonzero metric entropy.

MSC:
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37C45 Dimension theory of smooth dynamical systems
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