Regularity of the Anosov splitting and of horospheric foliations

Abstract

Abstract‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up toC2−ε. Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.

On contact Anosov flows

“…Unfortunately, C 1 strong stable and unstable foliations seem to be a quite rare phenomenon for higher dimensional Anosov flows [29], [10], [37]. One is therefore led to think that, unless some further geometrical structure is present, Anosov flows decay typically slower than exponentially.…”

Hyperbolic Dynamical Systems

“…The basic results from above apply essentially verbatim to the situation of a hyperbolic set other than the entire manifold. Notably Proposition 2.3.3 applies in that situation (the proof in [Hb3,Hb5] extends to that case). This observation can be strengthened in a remarkable way.…”

“…In that example the optimal Hölder exponent is 2/3 almost everywhere (with respect to volume) [A1]. One can make much stronger optimality assertions: w ¤ § ¥ § ¦ horospheric foliations [Hb3].…”

“…Zygmund holonomies) one may get non-Lipschitz holonomies (in higher dimension the Lipschitz property can fail even in negative curvature [Hb3]). If the curvature of a compact surface is negative except along a closed geodesic then the horocycle foliation may fail to be 1/2-Hölder at the corresponding orbit [GN] (this applies to subbundles as well as holonomies).…”

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

“…According to general regularity results, (P) implies that W u ǫ (x) and W s ǫ (x) are Lipschitz in x ∈ Λ. In fact, it follows from [Ha2] (see also [Ha1]) that assuming (P), the map Λ ∋ x → E u (x) is C 1+ǫ with ǫ = 2 inf x∈Λ (α x /β x ) − 1 > 0, in the sense that this map has a linearization at any x ∈ Λ that depends (uniformly Hölder) continuously on x. The same applies to the map Λ ∋ x → E s (x).…”