TY - JOUR

T1 - SMOOTH ERGODIC THEORY OF Zd-ACTIONS

AU - Brown, Aaron

AU - Rodriguez Hertz, Federico

AU - Wang, Zhiren

N1 - Publisher Copyright:
© 2023, American Institute of Mathematical Sciences. All rights reserved.

PY - 2023

Y1 - 2023

N2 - In the first part of this paper, we formulate a general setting in which to study the smooth ergodic theory of differentiable Zd-actions preserving a Borel probability measure. This framework includes actions by C1+Hölder diffeomorphisms of compact manifolds. We construct intermedi-ate unstable manifolds and coarse Lyapunov manifolds for the action as well as establish controls on their local geometry. In the second part, we consider the relationship between entropy, Lya-punov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier–Young entropy formulas. In the third part, for a smooth action of Zd preserving a Borel probability measure, we show that entropy satisfies a certain “product structure” along coarse unstable manifolds. Moreover, given two smooth Zd-actions— one of which is a measurable factor of the other—we show that all coarse-Lyapunov exponents contributing to the entropy of the factor system are coarse Lyapunov exponents of the total system. As a consequence, we derive an Abramov–Rohlin formula for entropy subordinated to coarse Lyapunov manifolds.

AB - In the first part of this paper, we formulate a general setting in which to study the smooth ergodic theory of differentiable Zd-actions preserving a Borel probability measure. This framework includes actions by C1+Hölder diffeomorphisms of compact manifolds. We construct intermedi-ate unstable manifolds and coarse Lyapunov manifolds for the action as well as establish controls on their local geometry. In the second part, we consider the relationship between entropy, Lya-punov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier–Young entropy formulas. In the third part, for a smooth action of Zd preserving a Borel probability measure, we show that entropy satisfies a certain “product structure” along coarse unstable manifolds. Moreover, given two smooth Zd-actions— one of which is a measurable factor of the other—we show that all coarse-Lyapunov exponents contributing to the entropy of the factor system are coarse Lyapunov exponents of the total system. As a consequence, we derive an Abramov–Rohlin formula for entropy subordinated to coarse Lyapunov manifolds.

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U2 - 10.3934/jmd.2023014

DO - 10.3934/jmd.2023014

M3 - Article

AN - SCOPUS:85161307385

SN - 1930-5311

VL - 19

SP - 455

EP - 540

JO - Journal of Modern Dynamics

JF - Journal of Modern Dynamics

ER -