Dimension of invariant measures for affine iterated function systems

2019/01/07 by De‐Jun Feng, Feng, De-Jun · 6 citations
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1901.01691

openalex publication_date 2019/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \S_i\_{i∈ Λ} be a finite contracting affine iterated function system (IFS) on \Bbb R^d. Let (Σ,σ) denote the two-sided full shift over the alphabet Λ, and π:Σ→ \Bbb R^d be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.

Citations

• Fractal Geometry: Mathematical Foundations and Applications.
• Products of Random Matrices
• Dimension, entropy and Lyapunov exponents

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