Related papers: A Denjoy counterexample for circle maps with an ha…

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by…

We give an example of a non-noetherian quasi-analytic ring constructed using a quasi-analytic Denjoy-Carleman class. If we denote by $ \mathcal{D}_n$ the ring of those $ C^\infty$ quasianalytic function germs at $0\in \mathbb{R}^n$ which…

Given any Liouville number $\alpha$, it is shown that various subspaces are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and…

We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.

In this work we present an example of C^\infty-diffeomorphism of a compact 4-manifold such that it admits a global SRB measure \mu but for which the special ergodic theorem doesn't hold. Namely, for this transformation there exist a…

Let $X\subset \mathbb{C}^n; Y\subset \mathbb{C}^m$ be closed affine varieties and let $\phi: X\to Y$ be an algebraic bi-Lipschitz homeomorphism. Then ${\rm deg}\ X={\rm deg}\ Y.$ Similarly, let $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset…

We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the…

Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas…

We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable in a…

If f is a bijection from C^n onto a complex manifold M, which conjugates every holomorphic map in C^n to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to…

We present a number of rigidity results concerning holomorphic dynamical systems admitting rotation quasicircles. Firstly, we show the absence of line fields on the Julia set of any rational map that is geometrically finite away from a…

We find conditions for two piecewise C^{2+\nu} homeomorphisms f and g of the circle to be C^1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary…

Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open…

Let $M$ be a smooth connected orientable closed surface and $f_0\in C^\infty(M)$ a function having only critical points of the $A_\mu$-types, $\mu\in\mathbb N$. Let ${\mathcal F}={\mathcal F}(f_0)$ be the set of functions $f\in C^\infty(M)$…

We characterize the simply connected domains $\Omega\subsetneq\mathbb{C}$ that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of $\Omega$ without fixed points has a Denjoy-Wolff point. We demonstrate that this…

We construct a $C^1$ symplectic twist map f of the annulus that has an essential invariant curve C such that: --C is not differentiable; -- the dynamic of f restricted to C is conjugated to the one of a Denjoy counter-example.

Fix an irrational number $\alpha$, and consider a random walk on the circle in which at each step one moves to $x+\alpha$ or $x-\alpha$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can…