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

DOI10.1007/BF01367764zbMath0916.28012arXivmath/9612223MaRDI QIDQ1273645

Nikita Sidorov, Anatoly M. Vershik

Publication date: 11 April 1999

Published in: Monatshefte für Mathematik (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9612223

zbMATH Keywords

entropyHausdorff dimensiongolden meanErdős measureFibonacci automorphismgoldenshift

Mathematics Subject Classification ID

Measure-preserving transformations (28D05) Entropy and other invariants (28D20) Low-dimensional dynamical systems (37E99)

Related Items (27)

Recent progress on Bernoulli convolutions ⋮ Universal \(\beta\)-expansions ⋮ On small bases which admit countably many expansions with multiple digits ⋮ On alpha-adic expansions in Pisot bases ⋮ The baker’s map with a convex hole ⋮ Bernoulli convolutions associated with certain non—Pisot numbers ⋮ Purity results for some arithmetically defined measures ⋮ Finite orbits in multivalued maps and Bernoulli convolutions ⋮ Growth rate for beta-expansions ⋮ Circular words and three applications: factors of the Fibonacci word, \(\mathcal F\)-adic numbers, and the sequence \(1, 5, 16, 45, 121, 320,\dots\) ⋮ The Erdős-Vershik problem for the golden ratio ⋮ A certain family of self-similar sets ⋮ On small bases which admit countably many expansions ⋮ Homoclinic processes and invariant measures for hyperbolic toral automorphisms ⋮ Harmonic analysis of iterated function systems with overlap ⋮ Denseness of intermediate $\beta $-shifts of finite-type ⋮ Critical bases for ternary alphabets ⋮ Cantor type functions in non-integer bases ⋮ Digit frequencies and Bernoulli convolutions ⋮ The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers ⋮ Random walks on \(\mathbb{Z}\) with exponentially increasing step length and Bernoulli convolutions ⋮ Open maps: small and large holes with unusual properties ⋮ Multifractal formalism for self-similar measures with weak separation condition ⋮ Branching random walk with exponentially decreasing steps, and stochastically self-similar measures ⋮ Expansions in non-integer bases: lower, middle and top orders ⋮ On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem ⋮ On the Gibbs properties of the Erdős measure

Cites Work

• Random walks on discrete groups: Boundary and entropy
• The natural extension of the \(\beta\)-transformation
• Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts
• The lattices of ideals of multizigzags and the enumeration of Fibonacci partitions
• A dimension formula for Bernoulli convolutions
• Bijective arithmetic codings of hyperbolic automorphisms of the \(2\)-torus, and binary quadratic forms
• Entropy and singularity of infinite convolutions
• Representations for real numbers and their ergodic properties
• On theβ-expansions of real numbers
• Dimension, entropy and Lyapunov exponents
• Representations of numbers and finite automata
• Arithmetic construction of sofic partitions of hyperbolic toral automorphisms
• Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution
• AUTOMATIC CONVERSION FROM FIBONACCI REPRESENTATION TO REPRESENTATION IN BASE φ, AND A GENERALIZATION
• ENTROPY, A COMPLETE METRIC INVARIANT FOR AUTOMORPHISMS OF THE TORUS
• Characterization of the unique expansions $1=\sum^{\infty}_{i=1}q^{-n_ i}$ and related problems
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