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Dynamical Borel-Cantelli lemmas for gibbs measures

Abstract

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA n ⊃ X and μ-almost every pointx∈X the inclusionT n x∈A n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

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Correspondence to N. Chernov.

Additional information

Partially supported by NSF grant DMS-9732728.

Partially supported by NSF grant DMS-9704489.

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Chernov, N., Kleinbock, D. Dynamical Borel-Cantelli lemmas for gibbs measures. Isr. J. Math. 122, 1–27 (2001). https://doi.org/10.1007/BF02809888

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Keywords

  • Periodic Orbit
  • Gibbs Measure
  • Left Endpoint
  • Markov Partition
  • Topological Pressure