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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.

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