Related papers: Bifurcations in the Space of Exponential Maps

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at…

We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values…

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this…

In this article, we define a locally finite graph $X$ as $\eta$-polynomially hyperbolic if there exists a Lipschitz map $\varphi : X \to Z$ to some hyperbolic space $Z$ satisfying the following condition: there exists $C \geq 0$ such that…

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period…

In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is…

Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary…

We show that the base space of a homotopy cofibration is locally hyperbolic under various conditions. In particular, if these manifolds admit a rationally elliptic closure, then almost all punctured manifolds and almost all manifolds with…

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same…

Let $\Omega$ be a domain with noncompact boundary. It is known that the Helmholtz decomposition is not always valid in $L^p(\Omega)$ except for the energy space $L^2 (\Omega)$. In this paper we consider a typical unbounded domain whose…

In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any…

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of…

Inspired by questions of convergence in continued fraction theory, Erd\H{o}s, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of M\"obius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with…

We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making…

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals…

Parabolic implosion describes the enrichment of Julia sets when a parabolic fixed point is perturbed. It is also natural to study parabolic implosion in parameter spaces. In particular, when one perturbs properly the family of cubic…

We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a…

According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by…

An important question is to describe topological conjugacy classes of dynamical systems. Here we show that within the space of real analytic one-dimensional maps with critical points of prescribed order, the conjugacy class of a map is a…

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…