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Looking to the fundamental domains of space groups we can investigate in which space they can be realized. If this space is hyperbolic, then the corresponding space group is also hyperbolic. In addition to the usual methods for…
We investigate some statistical and transport properties of the relativistic standard map. Through the Hamiltonian of a wave packet under an electric potential, we are able to obtain a relativistic version of the standard map, where there…
We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales H\"older continuous) regularity of such…
We study the $u$-Gibbs measures of a certain class of uniformly hyperbolic skew products on $\mathbb{T}^4$. These systems have a strong unstable and a weak unstable directions. We show that $C^r$-dense and $C^2$-open in this set every…
In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order…
There exists a $C^2$-open and $C^1$-dense subset of vector fields exhibiting singular-hyperbolic attracting sets (with codimension-two stable bundle), in any $d$-dimensional compact manifold ($d\ge3$), which mix exponentiallu with respect…
We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a…
We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the…
We study $C^1$-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. {In dimension $3$, an important class of…
In the 1960s and 1970s a large part of the theory of dynamical systems concerned the case of uniformly hyperbolic or Axiom A dynamical system and abstract ergodic theory of smooth dynamical systems. However since around 1980 an emphasize…
If $f:{\bf R}^d\to{\bf C}$ is bounded and $f$'s H\"older $\alpha$-modulus of continuity grows no faster than $(1+\vert x\vert)^M$ ($M\geq0$) then, for every $\epsilon>0$, there is a $\beta>0$ such that $f$'s H\"older $\beta$-modulus grows…
We investigate the decay rates of correlations for nonuniformly hyperbolic systems with or without singularities, on piecewise H\"older observables. By constructing a new scheme of coupling methods using the probability renewal theory, we…
Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when…
We construct symplectomorphisms in dimension $d\geq 4$ having a semi-local robustly transitive partially hyperbolic set containing $C^2$-robust homoclinic tangencies of any codimension $c$ with $0<c\leq d/2$.
In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in…
In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an HHG…
We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction,…
We discuss recent progress in understanding the dynamical properties of partially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are density of stable ergodicity and stable accessibility, center Lyapunov…
We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…
We consider H\"older continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $M$. We obtain several results for this setting. If a cocycle is bounded in…