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Tied-down occupation times of infinite ergodic transformations

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Abstract

We prove distributional limit theorems (conditional and integrated) for the occupation times of certain weakly mixing, pointwise dual ergodic transformations at “tied-down” times immediately after “excursions”. The limiting random variables include the local times of q-stable Lévy-bridges (1 < q ≤ 2) and the transformations involved exhibit “tied-down renewal mixing” properties which refine rational weak mixing. Periodic local limit theorems for Gibbs—Markov and AFU maps are also established.

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Acknowledgement

The authors are grateful to the anonymous referee for useful comments and in particular for raising the issue of remark 5.4.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Jon Aaronson

  2. Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka, 560-0043, Japan

    Toru Sera

Authors
  1. Jon Aaronson
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  2. Toru Sera
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Correspondence to Jon Aaronson.

Additional information

Happy birthday Benjy!

Dedicated to Benjamin Weiss on his 80th birthday

Aaronson’s research was partially supported by ISF grant No. 1289/17. Sera’s research was partially supported by JSPS KAKENHI grants No. 19J11798 and No. JP21J00015.

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Aaronson, J., Sera, T. Tied-down occupation times of infinite ergodic transformations. Isr. J. Math. 251, 3–47 (2022). https://doi.org/10.1007/s11856-022-2430-3

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  • DOI: https://doi.org/10.1007/s11856-022-2430-3

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