Related papers: Nonrigidity of piecewise-smooth circle maps

We study the asymptotical behaviour of iterates of piecewise contractive maps of the interval. It is known that Poincar\'e first return maps induced by some Cherry flows on transverse intervals are, up to topological conjugacy, piecewise…

Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown…

We consider families of dynamics that can be described in terms of Perron-Frobenius operators with exponential mixing properties. For piecewise C^2 expanding interval maps we rigorously prove continuity properties of the drift J(l) and of…

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent…

Let $M$ be a connected 1-manifold, i.e., $M = \R \cong (0, 1), [0, 1), [0, 1]$, or $S^1$, and let $\Homeo_+(M)$ (resp. $\Diff_+^1(M)$) be the group of orientation-preserving homeomorphisms (resp. $C^1$ diffeomorphisms) of $M$. It is a…

We consider two transitive $3$-dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with $C^1$ regular stable or unstable…

We classify the non-negative critical points in $W^{1,p}_0(\Omega)$ of \[ J(v)=\int_\Omega H(Dv)-F(x, v)\, dx \] where $H$ is convex and positively $p$-homogeneous, while $t\mapsto \partial_tF(x, t)/t^{p-1}$ is non-increasing. Since $H$ may…

We study a simple problem that arises from the study of Lorentz surfaces and Anosov flows. For a non decreasing map of degree one $h:\mathbb{S}^1\to \mathbb{S}^1$, we are interested in groups of circle diffeomorphisms that act on the…

Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…

We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed…

Let $f$ be an orientation preserving homeomorphism of $S^2$ which has a (nontrivial) continuum $X$ as a minimal set. Then there are exactly two connected components of $S^2\setminus X$ which are left invariant by $f$ and all the others are…

We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the…

Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $\mu_t$ is the corresponding SRB measure for $f_t$, we study the regularity of…

We prove that any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a $C^{1+\alpha}$ circle diffeomorphism, for some universal $\alpha>0$.

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown…

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for…

We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a M\"obius band, the two-point finite…

Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[…

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space…

We prove some basic results for a dynamical system given by a piecewise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the…