SMOOTH ERGODIC THEORY OF Z^d-ACTIONS

In the first part of this paper, we formulate a general setting in which to study the smooth ergodic theory of differentiable Z^d-actions preserving a Borel probability measure. This framework includes actions by C^1+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 Z^d preserving a Borel probability measure, we show that entropy satisfies a certain “product structure” along coarse unstable manifolds. Moreover, given two smooth Z^d-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.