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We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical…

We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full…

Commuting maps on a class of algebras called inflated algebras are investigated. In particular, we can prove that every commuting map $\theta$ on such an algebra is of the form $\theta(x)=c x+\mu(x)$, where $c$ belongs to the base field $K$…

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the…

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in…

We describe the topological behavior of typical orbits of complex quadratic polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type. Here we prove that for such Brjuno values of alpha the closure of the critical orbit,…

Given an ergodic measure with positive entropy and only positive Lyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps…

We show that every point $x_0\in [0,1]$ carries a representation of a $C^*$-algebra that encodes the orbit structure of the linear mod 1 interval map $f_{\beta,\alpha}(x)=\beta x +\alpha$. Such $C^*$-algebra is generated by partial…

We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate…

From previous work, we know how to obtain type II_0 E_0-semigroups using boundary weight doubles (\phi, \nu), where \phi: M_n(\C) \to M_n(\C) is a unital q-positive map and \nu is a normalized unbounded boundary weight over L^2(0, \infty).…

Let $\rho_\Sigma=h(|z|^2)$ be a metric in a Riemann surface $\Sigma$, where $h$ is a positive real function. Let $\mathcal H_{r_1}=\{w=f(z)\}$ be the family of univalent $\rho_\Sigma$ harmonic mapping of the Euclidean annulus…

We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise…

We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the…

We investigate the critical points of the basic (quasi-)modular forms $E_2$, $E_4$, and $E_6$. They occur where some associated polymorphic functions have poles. By an explicit description of these polymorphic functions as conformal maps,…

We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables…

We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of…

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…

Let $D$ be the set of $\beta \in (1, 2]$ such that $f_\beta$ is a symmetric tent map with finite critical orbit. For $\beta \in D$, by operating Denjoy like surgery on $f_{\beta}$, we constructed a $C^1$ unimodal map $\tilde{g}_\beta$…

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$, we consider the problem of restoring the metric $g$ and the magnetic potential $\alpha$ from the values of the Ma\~n\'e action potential between…