Related papers: Rigidity of analytic and smooth bi-cubic multicrit…

We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every…

We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.

Classical results by Poincar\'e and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of…

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…

In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary…

We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are…

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with…

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity…

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit…

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like…

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive…

For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…

We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove…

We consider infinitely renormalizable unimodal mappings with topological type which is periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point…

This paper is a step towards the complete topological classification of {\Omega}-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically…

This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary,…

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a…

In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e.,…

Assume that there exists a smooth map between two closed manifolds $M^m\to N^k$ with only finitely many cone-like singular points, where $2\leq k\leq m\leq 2k-1$. If $(m,k)\not\in\{(2,2), (4,3), (5,3), (8,5), (16,9)\}$, then $M^m$ admits a…