Related papers: Example of a diffeomorphism for which the special …

Suppose f is a C^{1+\epsilon} surface diffeomorphism with positive topological entropy. For every positive \delta strictly smaller than the topological entropy of f we construct an invariant Borel set E such that (a) f|E has a countable…

This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $\rho$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $\rho$. We propose that…

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\Sigma,\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field…

We give a sufficient condition for the ergodicity of the Lebesgue measure for an iterated function system of diffeomorphisms. This is done via the induced iterated function system on the space of continuum (which is called hyper-space). We…

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2…

In this paper, we study the differentiability of SRB measures for partially hyperbolic systems. We show that for any $s \geq 1$, for any integer $\ell \geq 2$, any sufficiently large $r$, any $\varphi \in C^{r}(\T, \R)$ such that the map $f…

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full…

Let Aff(X) be the group of affine diffeomorphisms of a closed homogeneous manifold X=G/B admitting a G-invariant Lebesgue-Haar probability measure $\mu$. For $f_0\in$ Aff(X), let $Z^\infty(f_0)$ be the group of $C^\infty$ diffeomorphisms of…

In this article we construct two minimal unique ergodic diffeomorphisms $\alpha$ and $\beta$ on $S^3 \times S^{6} \times S^{8} $. We will show that $C(S^3 \times S^{6} \times S^{8}) \rtimes_\alpha \mathbb{Z} $ and $C(S^3 \times S^{6} \times…

We show that any area-preserving $C^r$-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be $C^r$-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every $1\le r\le \infty$.…

Birkhoff's theorem is a classic result that characterizes locally spherically symmetric solutions of the Einstein equations. In this paper, we illustrate the consequences of its local nature for the cases of vacuum and positive cosmological…

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface…

Let b be an integer and mu a probability measure on [0,1] which is invariant and ergodic multiplication by b mod 1, and 0<dim(mu)<1. Let f be a diffeomorphism between open subsets of the line. We show that if the measures mu and f(mu) are…

In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_{\Lambda}M = E\oplus F$ on an…

For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},\mu,T)$ we prove that for any $f,g\in L^{\infty}_{\mu}(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor…

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in…

We construct, for $p>n$, a concrete example of a complete non-compact $n$-dimensional Riemannian manifold of positive sectional curvature which does not support any $L^p$-Calder\'on-Zygmund inequality: \[ \forall\,\varphi\in…

We present a simple, computation free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold which is invariant and normally hyperbolic persists under small perturbations of the…

In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar…

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This…