Related papers: Maximal pseudometrics and distortion of circle dif…
We provide a new angle and obtain new results on a class of metrics on length-normalized curves in $d$ dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the…
Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group…
The group $\text{Diff}(\mathcal{M})$ of diffeomorphisms of a closed manifold $\mathcal{M}$ is naturally equipped with various right-invariant Sobolev norms $W^{s,p}$. Recent work showed that for sufficiently weak norms, the geodesic…
We study renormalizations of piecewise smooth homeomorphisms on the circle, by considering such maps as generalized interval exchange maps of genus one. Suppose that $Df$ is absolutely continuous on each interval of continuity and…
Let ${\cal F}\_\lambda(S^1)$ be the space of tensor densities of degree (or weight) $\lambda$ on the circle $S^1$. The space ${\cal D}^k\_{\lambda,\mu}(S^1)$ of $k$-th order linear differential operators from ${\cal F}\_\lambda(S^1)$ to…
The recent discovery of metadislocations in some periodic complex metallic alloys and of their 'phason' defects has given a new impetus to the study of QC approximant defects. In this paper we emphasize: 1- that approximants differ from a…
We prove that every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the…
Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X_0 and X_1 are bi-Hamiltonian relatively to a modified Lie-Poisson…
We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of…
A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$ has a hyperbolic set $\La_g$, depending continuously on $g$, such that the stable and…
We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing…
$C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization.…
In this paper we consider a non-atomic invariant hyperbolic measure $\mu$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $\mu$ a.e. points. We show an…
We show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of…
We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a…
In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those homotopic to Anosov (or Derived from Anosov diffeomorphisms), diffeomorphisms in neighborhoods of the…
In this paper, we give a characterization for any abelian subgroup G of a lie group of diffeomorphisms maps of C^n, having a somewhere dense orbit G(x), x in C^n: G(x) is somewhere dense in C^n if and only if there are f_{1},....,f_{2n+1 in…
What kind of dynamics do we observe in general on the circle? It depends somehow on the interpretation of "in general". Everything is quite well understood in the topological (Baire) setting, but what about the probabilistic sense? The main…
We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types,…
Let $E$ be a finite-dimensional real vector space and $M\subseteq E$ be a convex polytope with non-empty interior. We turn the group of all $C^\infty$-diffeomorphisms of $M$ into a regular Lie group.