On the L^q dimension of stationary measures for Möbius iterated function systems

2025/01/23 by Shunsuke Usuki, Usuki, Shunsuke · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2501.13729

openalex publication_date 2025/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the L^q dimension D(ν,q) (q>1) of stationary measures ν for Möbius iterated function systems on ℝ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]Shm19. As the result, we show that there is the dichotomy: the L^q spectrum τ(ν,q)=(q-1)D(ν,q) is equal to the desired value min\\widetildeτ(ν,q),q-1\ for any q>1, where \widetildeτ(ν,q) is the zero of the canonical pressure function, or there exist q₀>1 and 0<α<1 such that τ(ν,q)=min\\widetildeτ(ν,q),q-1\ for 1

Cited by

• On exponential separation of analytic self-conformal sets on the real line

Related

• Dimension bounds for invariant measures of bi-Lipschitz iterated function systems
• Dimension estimates for invariant measures of contracting-on-average iterated function systems
• On the dimension of self-similar measures with complicated overlaps
• A strongly irreducible affine iterated function system with two invariant measures of maximal dimension
• Special families of piecewise linear iterated function systems