Related papers: Rigidity of $C^2$ Infinitely Renormalizable Unimod…

We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll}…

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*-algebra, there is a corresponding *-automorphism \alpha of another unital C*-algebra such that the two sequences P, P^2,P^3,... and \alpha,…

In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval), 2) for which $\omega(c)$ is a Cantor…

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(\alpha,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the…

We further develop the relationship between $\beta$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^{\alpha}_{\mu;2}(x,r)$ at…

We prove that if $G\subset\text{Diff}^{1}(\mathbb{R}^2)$ is an Abelian subgroup generated by a family of commuting diffeomorphisms of the plane, all of which are $C^{1}$-close to the identity in the strong $C^{1}$-topology, and if there…

We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $\Omega$ and a base invertible ergodic map…

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…

Finiteness conditions for $C^*$-algebras like AF-embeddability, quasidiagonality, stable finiteness have been studied by many authors and shown to be equivalent for certain classes of $C^*$-algebras. For example, Schfhauser proves that…

We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_\zeta$ that contains a Brjuno rotation $f_0(z)=z+\alpha$, all maps…

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps…

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(\lambda)$ for $\lambda$ in $\mathbb{C}$, defined by the condition that each $f\in…

A semigroup (dynamical system) generated by $C^{1+\alpha}$-contracting mappings is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives…

In this note, we prove a rigidity result for proper holomorphic maps between unit balls that have many symmetries and which extend to $\mathcal{C}^2$-smooth maps on the boundary.

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of…

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

We prove the a priori bounds for infinitely renormalizable quadratic polynomials for which we can find an infinite sequence of primitive renormalizations such that the ratios of the periods of successive renormalizations is bounded. This…

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge…

We study the infinitesimal rigidity of equivariant minimal maps from the universal cover of a smooth oriented surface (possibly non-compact) into a Riemannian symmetric space, focusing on representations arising from cyclic harmonic…

We show that a finite group $G$ admitting an automorphism $\alpha$ such that the function $G\rightarrow G$, $g\mapsto g\alpha(g)$, is bijective is necessarily solvable.