A+ CATEGORY SCIENTIFIC UNIT

Random Lochs’ Theorem

Charlene Kalle, Evgeny Verbitskiy, Benthen Zeegers Studia Mathematica MSC: Primary 11K55; Secondary 28D20, 37A10, 60F05, 11K60, 37H15, 37A44, 11J83. DOI: 10.4064/sm211028-24-2 Published online: 7 June 2022

Abstract

In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number $x$ that can be determined from just knowing its first $n$ decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs’ Theorem for a broad class of random dynamical systems, and under additional assumptions we prove a corresponding Central Limit Theorem as well. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin’s formula for entropy.

Authors

  • Charlene KalleMathematisch Instituut
    Leiden University
    Niels Bohrweg 1
    2333 CA Leiden, The Netherlands
    e-mail
  • Evgeny VerbitskiyMathematisch Instituut
    Leiden University
    Niels Bohrweg 1
    2333 CA Leiden, The Netherlands
    and
    Bernoulli Institute
    Groningen University
    Nijenborgh 9
    9747 AG Groningen, The Netherlands
    e-mail
  • Benthen ZeegersMathematisch Instituut
    Leiden University
    Niels Bohrweg 1
    2333 CA Leiden, The Netherlands
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image