Finiteness of ergodic measures of maximal entropy for certain skew product systems
Abstract
Consider a C1+β skew product F over a topologically mixing Anosov diffeomorphism f, with fiber dynamics driven by a C1+β ℝk-action {ρbf t} on a closed manifold Y through a C1+β cocycle τ. Suppose that the fiber action preserves an ergodic smooth measure of maximal entropy mMME. Under a nondegeneracy condition called joint nontriviality, and assuming that each element ρbf t is either of class C∞ or hyperbolic, we prove that F admits only finitely many ergodic measures of maximal entropy. Moreover, every such measure is of the form μ× mMME, where μ is an ergodic equilibrium state for a Hölder potential on the base. If, in addition, the fiber action is exponentially mixing of all orders, then every ergodic MME is Bernoulli, exponentially mixing of all orders, and satisfies the Central Limit Theorem. Our main novelties are entropy formulas for skew products, obtained through the cone property and r-neutralized entropy, and a relative Ledrappier--Young theorem that converts the fiber entropy contribution into absolute continuity of fiberwise conditional measures. Combined with nonlinear thermodynamic formalism, rotation theory and Yomdin theory, these yield the finiteness of the base projections and the product rigidity of ergodic MMEs.
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BibTeX
@misc{dong2026finiteness,
title = {{Finiteness of ergodic measures of maximal entropy for certain skew product systems}},
author = {Changguang Dong and Qiujie Qiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34209},
eprint = {2610.34209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34209},
}
RIS
TY - UNPB
TI - Finiteness of ergodic measures of maximal entropy for certain skew product systems
AU - Changguang Dong
AU - Qiujie Qiao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34209
UR - https://arxiv.org/abs/2610.34209
AB - Consider a C^1+β skew product F over a topologically mixing Anosov diffeomorphism f, with fiber dynamics driven by a C^1+β ℝ^k-action {ρ_bf t} on a closed manifold Y through a C^1+β cocycle τ. Suppose that the fiber action preserves an ergodic smooth measure of maximal entropy m_MME. Under a nondegeneracy condition called joint nontriviality, and assuming that each element ρ_bf t is either of class C^∞ or hyperbolic, we prove that F admits only finitely many ergodic measures of maximal entropy. Moreover, every such measure is of the form μ× m_MME, where μ is an ergodic equilibrium state for a Hölder potential on the base. If, in addition, the fiber action is exponentially mixing of all orders, then every ergodic MME is Bernoulli, exponentially mixing of all orders, and satisfies the Central Limit Theorem. Our main novelties are entropy formulas for skew products, obtained through the cone property and r-neutralized entropy, and a relative Ledrappier--Young theorem that converts the fiber entropy contribution into absolute continuity of fiberwise conditional measures. Combined with nonlinear thermodynamic formalism, rotation theory and Yomdin theory, these yield the finiteness of the base projections and the product rigidity of ergodic MMEs.
ER -