On the timescale at which statistical stability breaks down

  • Neil Dobbs

    University College Dublin, Ireland
  • Alexey Korepanov

    Exeter University, UK
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Abstract

In dynamical systems, understanding statistical properties shared by most orbits and how these properties depend on the system are basic and important questions. Statistical properties may persist as one perturbs the system (statistical stability is said to hold), or may vary wildly. The latter case is our subject of interest, and we ask at what timescale does statistical stability break down. This is the time needed to observe, with a certain probability, a substantial difference in the statistical properties as described by (large but finite time) Birkhoff averages.

The quadratic (or logistic) family is a natural and fundamental example where statistical stability does not hold. We study this family. When the base parameter is of Misiurewicz type, we show, sharply, that if the parameter changes by , it is necessary and sufficient to observe the system for a time at least of the order of to see the lack of statistical stability.

Cite this article

Neil Dobbs, Alexey Korepanov, On the timescale at which statistical stability breaks down. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 175–199

DOI 10.1016/J.ANIHPC.2020.06.001