[Paper Review] Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems

Nikita Sidorov, A. M. Vershik|ArXiv.org|Dec 15, 1996

Mathematical Dynamics and Fractals11 references11 citations

TL;DR

This paper introduces a two-sided Erdös measure on the 2-torus using the golden ratio, constructs the goldenshift transformation preserving both Erdös and Lebesgue measures, and proves it is Bernoulli. This yields explicit formulas for the entropy dimension of Erdös measure on the interval and resolves related problems in symbolic dynamics and number theory.

ABSTRACT

We define a two-sided analog of 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 $\Bbb T^2$ which 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.

Motivation & Objective

• To define a two-sided analog of Erdös measure on the 2-torus using the golden ratio expansion.
• To construct the goldenshift transformation as an induced automorphism of the Fibonacci toral automorphism.
• To prove that the goldenshift is Bernoulli with respect to both Erdös and Lebesgue measures.
• To derive explicit formulas for the entropy dimension of Erdös measure on the interval.
• To analyze central measures on the Fibonacci graph and their dynamical implications.

Proposed method

• Define the two-sided Erdös measure on the 2-torus via expansions in powers of the golden ratio.
• Construct the goldenshift as the induced transformation of the standard shift on symbolic sequences with Fibonacci constraints.
• Prove that the goldenshift preserves both Erdös and Lebesgue measures on the 2-torus.
• Establish the Bernoulli property of the goldenshift using spectral and ergodic-theoretic techniques.
• Use the Bernoulli nature of the goldenshift to compute the entropy dimension of the Erdös measure via measure-theoretic entropy.
• Analyze central measures on the Fibonacci graph using the dynamics of the goldenshift and its invariant measures.

Experimental results

Research questions

• RQ1 What are the ergodic properties of the two-sided Erdös measure on the 2-torus?
• RQ2 Is the goldenshift transformation Bernoulli with respect to both Erdös and Lebesgue measures?
• RQ3 What is the entropy dimension of the Erdös measure on the unit interval?
• RQ4 How do central measures on the Fibonacci graph relate to the dynamics of the goldenshift?
• RQ5 What is the relationship between the goldenshift and the Fibonacci toral automorphism?

Key findings

• The goldenshift transformation is proven to be Bernoulli with respect to both the Erdös and Lebesgue measures on the 2-torus.
• The entropy dimension of the Erdös measure on the interval is computed explicitly using the dynamics of the goldenshift.
• The paper provides a direct derivation of the entropy in the sense of Garsia-Alexander-Zagier for the Erdös measure.
• The construction establishes a natural link between symbolic dynamics on the Fibonacci graph and the 2-torus via the goldenshift.
• The goldenshift arises as the induced automorphism of the standard shift under the Fibonacci substitution rule.
• The study confirms that the Erdös measure is not a Gibbs measure but still admits a well-defined entropy dimension through the goldenshift dynamics.