The Metric Entropy of Diffeomorphisms: Part II: Relations between Entropy, Exponents and Dimension

Abstract

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Extremal Lyapunov exponents: an invariance principle and applications

“…This yields (1) and (2). Part (3) is a direct consequence of (2) and the definition (35), in the case y ∈ W s ε (y). The general statement follows, using the invariance property (sh2): [43,Section 4] where stronger results are proved in detail using similar methods, in the context of linear cocycles.…”

“…When the vectors in the center bundle E c also have non-zero Lyapunov exponents, that is, when lim 1 n log Df n (x)v = 0 ∀v ∈ E c x at typical points x ∈ M , one can build on (non-uniform) hyperbolicity theory to derive important geometric and statistical information on the dynamics. See Pesin [38,39], Ledrappier, Young [32,34,35], Katok [29], Barreira, Pesin, Schmeling [8], Young [45,46], and Alves, Bonatti, Viana [2,15].…”

The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture

“…, j), the conditional measures µ ij x are trivial. This is a (slight adaptation of a) special case of the "Ledrappier-Young entropy formula" [12,13] and is of great importance, for the Theorems 5.1 and 5.2 are not useful without a reasonable way to verify the conditions on µ ij x . The above result provides this, and, indeed, in many applications it is Theorem 4.3 rather than Theorem 5.2 which is directly applicable.…”

“…Now let us briefly -and very heuristically -indicate how one might use Lemma 5.1. The assertion 5.1 says, in particular, that the value of x → µ 13 x is "the same" (at least, proportional) at x and at n 12 (t)x, except for a set of t of µ 12…”

Rigidity of measures—The high entropy case and non-commuting foliations

“…We call the expression vol µ (α m , H, x) volume decay entropy at x -it can be thought of as a combination of the dimension of µ H x at x and the contraction rates of θ m (see Ledrappier-Young's entropy formula [20]). In case P x is invariant under θ, mod(α m , P x ) is the negative logarithm of the module of the restriction of θ m to P x .…”