Related papers: Period doubling, entropy, and renormalization

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…

Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of $\pro$ of degree 2, defined over $K$,…

We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form $\int_\Omega f(|\mathrm{D} u|)\,\mathrm{d} x$. The analysis…

Two classes of time-periodic systems of ordinary differential equations with a small nonnegative parameter, those with fast and slow time, are studied. Right-hand sides of these systems are three times continuously differentiable with…

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

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…

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 study decomposition into simple arcs (i. e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph $G$ is a knotted loop, then there…

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}…

Some results of numerical study of the canonical map with a sawtooth force are given and discovered new unexpected dynamical effects are described. In particular, it is shown that if the values of the system parameter K belong to the…

Let f be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in the boundary of U, under certain hypothesis on…

We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: \begin{align*} f_{a,b}(x)=\left(2x+a+\dfrac{b}{\pi} \sin(2\pi x)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0…

We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as `strangely…

We continue the study of straightening maps for the family of polynomials of degree $d \ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and…

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable…

We study highly dissipative H\'enon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that…

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of…

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space…

The past decades have witnessed a transformation in characterizing condensed matter systems with topology. Aided by a refined understanding of topological band structures with crystalline symmetries that has emerged recently, many…