Monatshefte für Mathematik

, Volume 126, Issue 3, pp 215–261 | Cite as

Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

  • Nikita Sidorov
  • Anatoly Vershik
Article

Abstract

We define a two-sided analog of the Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on\(\mathbb{T}^2 \) that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.

1991 Mathematics Subject Classification

28D05 28D20 58F03 

Key words

Erdoes measure goldenshift Fibonacci automorphism Hausdorff dimension Fibonacci graph 

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Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Nikita Sidorov
    • 1
  • Anatoly Vershik
    • 1
  1. 1.St. Petersburg Department of Steklov Mathematical InstituteSt. PetersburgRussia

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