PreprintCC BY 4.0arXiv
Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative
Abstract
We prove exact dimensionality of ergodic stationary measures for random compositions of C1 diffeomorphisms of a closed manifold M that contract at a single Lyapunov scale. More precisely, let ν be a Borel probability measure on Diff1(M) such that logsupx|Dxf| and logsupx|Dxf−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 dim(μ)=hμF(ν)/(−λ), where hμF(ν) is the Furstenberg entropy.
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BibTeX
@misc{mukherjee2026exact,
title = {{Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative}},
author = {Subhasish Mukherjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10538},
eprint = {2609.10538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10538},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative
AU - Subhasish Mukherjee
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.10538
UR - https://arxiv.org/abs/2609.10538
AB - 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 Diff^1(M) such that logsup_x|D_xf| and logsup_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 dim(μ)=h_μ^F(ν)/(−λ), where h_μ^F(ν) is the Furstenberg entropy.
ER -