We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal in general, and exactly one the topologically transitive case. This answers a question Newhouse, who proved such always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, introduce homoclinic cla...

Let G be a finitely generated group which is hyperbolic relative to a finite family fH1; . . . ;Hng of subgroups. We prove that G is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space.

The fractal conservation law ∂tu + ∂x( f (u)) + (−∆)α/2u = 0 changes characteristics as α → 2 from non-local and weakly diffusive to local and strongly diffusive. In this paper we present a corrected finite difference quadrature method for (−∆)α/2 with α ∈ [0,2], combined with usual finite volume methods for the hyperbolic term, that automatically adjusts to this change and is uniformly converg...

We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

In this paper we extend to UCW-hyperbolic spaces the quantitative asymptotic regularity results for alternating Halpern–Mann iteration obtained by Dinis and second author CAT(0) spaces. These are new even uniformly convex normed Furthermore, a particular choice of parameter sequences, compute linear rates in W-hyperbolic quadratic T- U-asymptotic

An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C singularhyperbolic attractors, for some α > 0, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular...

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses required to understand the dynamics. highly influential hypothesis Gallavotti Cohen states large-scale effectively uniformly hyperbolic, which implies felicitous statistical properties. We obtain direct reliable numerical evidence, contrary hypothesis, existenc...

We consider continuous SL(2,R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hype...