Related papers: Two bifurcation sets of expansive Lorenz maps with…

Let $f$ be an expansive Lorenz map and $c$ be the critical point. The survivor set is denoted as $S_{f}(H):=\{x\in[0,1]: f^{n}(x)\notin H, \forall n\geq 0\}$, where $H$ is a open subinterval. Here we study the hole $H=(a,b)$ with $a\leq c…

Let $0<a<b<1$ and let $T$ be the doubling map. Set $\mathcal J(a,b):=\{x\in[0,1] : T^nx\notin (a,b), n\ge0\}$. In this paper we completely characterize the holes $(a,b)$ for which any of the following scenarios holds: {enumerate} $\mathcal…

For a map $S:X\to X$ and an open connected set ($=$ a hole) $H\subset X$ we define $\mathcal J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids $H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such that…

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but…

Hopf bifurcations have been studied perturbatively under two broad headings, viz., super-critical and sub-critical. The criteria for occurrences of such bifurcations have been investigated using the renormalization group. The procedure has…

Given $\beta\in(1,2],$ the $\beta$-transformation $T_\beta: x\mapsto \beta x\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set \[ \mathcal…

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…

We consider order preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(\ell_1, \ell_2)$. We detect a change in the geometry of the system. For $(\ell_1, \ell_2) \in [1,2]^2$ the geometry is…

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any…

We consider the conservative H\'enon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that $\lambda_+$ is the…

A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the…

Given $\beta>1$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit…

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth…

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of…

Let $F=(F_1, F_2, ..., F_m): \mathbb{C}^n \to \mathbb{C}^m$ be a polynomial dominant mapping with $n>m$. In this paper we give the relations between the bifurcation set of $F$ and the set of values where $F$ is not M-tame as well as the set…

We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic, i.e., for which the critical point $0$ is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric $\rho(z)…

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in…

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large…

We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-\Delta u+V(x)u=\left(I_\alpha\ast |u|^p\right)|u|^{p-2}u+\lambda u\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right.…

Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the…