Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative
Year: 2026
Type: preprint
Abstract: We prove exact dimensionality of ergodic stationary measures for random compositions of $C^1$ diffeomorphisms of a closed manifold $M$ that contract at a single Lyapunov scale. More precisely, let $ν$ be a Borel probability measure on $\mathrm{Diff}^1(M)$ such that $\log\sup_{x}\|{D_xf}\|$ and $\log\sup_{x}\|{D_xf^{-1}}\|$ are $ν$-integrable, and let $μ$ be an ergodic $ν$-stationary probability measure. If the top and bottom Lyapunov exponents of the derivative cocycle coincide and equal $λ<0$, then $μ$ is exact dimensional and $\mathrm{dim}(μ)=h_μ^{\mathrm{F}}(ν)/(-λ)$, where $h_μ^{\mathrm{F}}(ν)$ is the Furstenberg entropy.
Source: arXiv (Cornell University)
Author: Mukherjee, Subhasish
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212120435