Lyapunov exponents, entropy and periodic orbits for diffeomorphisms

Abstract

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Extremal Lyapunov exponents: an invariance principle and applications

“…Now, consider any countable set {x n } ⊂ supp µ such that the balls of radius δ/2 around these x n cover the support of µ. For each n, let B n be a zero µ-measure subset of the ball V (x n ) of radius δ around x n such that (29) m w = (h s z,w ) * m z for all z, w ∈ V (x n ) \ B n in the same stable piece. Let E be the set of all point x ∈ supp µ whose orbits never meet ∪ n B n and such that (30) m f l (x) = (F l x ) * m x for all l ∈ Z.…”

“…So, by Proposition 4.4, the two points z = f k (x) and w = f k (y) belong to the same stable piece (associated to x n ). Since they are outside B n , we may combine (29) and (30) with property (sh2) of s-holonomies to conclude that m y = (h s x,y ) * m x . This proves that the disintegration of m is essentially s-invariant, as claimed.…”

“…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].…”

Differentiability, rigidity and Godbillon-Vey classes for Anosov flows

“…Recall that the entropy of an Anosov flow is a measure of the growth rate of the lengths of the closed orbits of the flow [38]. Comparing the formulas (19) and (20) suggests that the term Def (g) should be viewed as a type of "mean variation" of the distribution of the closed orbits.…”

Variational equalities of entropy in nonuniformly hyperbolic systems

“…For each ergodic measure ν, we use Katok's definition of metric entropy( see [17]). For x, y ∈ M and n ∈ N, let…”