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The metric entropy of diffeomorphisms. I: Characterization of measures satisfying Pesin’s entropy formula. (English) Zbl 0605.58028

This is a joint review of Part I and Part II: Relations between entropy, exponents and dimension [ibid. 122, No. 3, 540–574 (1985, Zbl 1371.37012)].
These two papers are a major contribution to the smooth ergodic theory. The authors work with arbitrary \(C^2\)-diffeomorphisms of a compact Riemannian manifold. In the first part they characterize measures satisfying Pesin’s entropy formula as those which have absolutely continuous conditional measures on unstable manifolds.
In the second part they relate metric entropy, Lyapunov exponents and dimension in the following formula: \(h_m(f)=\int \sum_{i}\lambda^+_i\gamma_ i \,dm\) where \(\gamma_i\) is the dimension of \(m\) in the direction in which the Lyapunov exponent is \(\lambda_i\).
Reviewer: Maciej Wojtkowski

MSC:

37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

Citations:

Zbl 1371.37012
Full Text: DOI