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The recurrence dimension for piecewise monotonic maps of the interval. (English) Zbl 1170.37316
Summary: We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval \([0,1]\), giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.
MSC:
37E05 Dynamical systems involving maps of the interval
37C45 Dimension theory of smooth dynamical systems
28A80 Fractals
37B40 Topological entropy
28A78 Hausdorff and packing measures
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