相关论文: Monodromy through bifurcation locus of the Mandelb…

We study the dynamics of template iterations, consisting of arbitrary compositions of functions chosen from a finite set of polynomials. In particular, we focus on templates using complex unicritical maps in the family $\{ z^d + c, c \in…

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We…

We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…

In this paper we continue the study, began in \cite{CJK2}, of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values and an attracting fixed point. If we fix the multiplier of the fixed…

We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification…

We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the…

We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler…

For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of…

The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest are investigated, and the…

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are…

For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the…

We give an example of a family of endomorphisms of $\mathbb P^2 (\mathbb C)$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non empty interior.

Let $(X,x)$ be an isolated complete intersection singularity and let $f : (X,x) \to (\CC,0)$ be the germ of an analytic function with an isolated singularity at $x$. An important topological invariant in this situation is the…

The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to…

We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$…

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as…

We consider the monodromy at infinity and the monodromies around the bifurcation points of polynomial functions $f : \CC^n \longrightarrow \CC$ which are not tame and might have non-isolated singularities. Our description of their Jordan…

We study asymptotic dynamics in networks of coupled quadratic nodes. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in a network,…

While the forward trajectory of a point in a discrete dynamical system is always unique, in general a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through $x$ was called by…

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the…