Related papers: The rigidity problem for analytic critical circle …

We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism $L$ and a $C^\infty$ diffeomorphism $f$ of the torus. For a very weakly irreducible hyperbolic automorphism $L$ we show that any $C^1$…

It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical…

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space.…

Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be…

Let R be rational map. Critical point c is called summable if series $\sum_i\frac{1}{(R^i)'(R(c))}$ is absolutely convergent. Under some topological condition on postcritical set we prove that R can not be structurally stable if summable…

An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between…

We use a novel real-time formulation of the functional renormalization group (FRG) for dynamical systems with reversible mode couplings to study Model H, the conjectured dynamic universality class of the QCD critical point. We emphasize the…

We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family $P_c: x\mapsto x^2+c$ has…

The critical behavior for intermittency is studied in two coupled one-dimensional (1D) maps. We find two fixed maps of an approximate renormalization operator in the space of coupled maps. Each fixed map has a common relavant eigenvaule…

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p,$ whose post-critical set is finite in any bounded domain of $\mathbb{C}.$ In spite of being rigid on the sphere, such…

In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps, \textit{constant periodic data} imply \textit{same periodic data of these maps and their…

We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,\tau}$ action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $\tau>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the…

We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann…

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the…

The critical loci of a map $f:X\to Y$ between smooth schemes over a field $k$ are the locally closed subschemes $\Sigma^i(f)\subseteq X$ where the differential of $f$ has constant rank. We prove that if $f : X\to \mathbb A^r$ is the general…

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points…

We study the dynamics of the renormalization operator acting on the space of pairs (v,t), where v is a diffeomorphism and t belongs to [0,1], interpreted as unimodal maps x-->v(q_t(x)), where q_t(x)=-2t|x|^a+2t-1. We prove the so called…

We give a new proof of the result that if f and g are transcendental entire functions, then the composite function f(g) has infinitely many fixed points. The method yields a number of generalization of this result. In particular, it extends…

We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.

We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global Hyperbolicity of Renormalization conjecture for unimodal maps of bounded type with a critical exponent which is…