Let (f_λ) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold Λ. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on Λ, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).
In the study of global bifurcations of vector fields on $S^2$, it is important to distinguish a set "where the bifurcation actually occurs", -- the bifurcation support. Hopefully, it is sufficient to study the bifurcation in a neighborhood of the support only. The first definition of bifurcation support was proposed by V.Arnold. However this set appears to be too small. In particular, the newly discovered effect, an open domain in the space of three-parametric families on $S^2$ with no structurally stable families, is not visible in a neighborhood of the bifurcation support. In this article, w…
We prove that given any $θ_1,\ldots,θ_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of respective multipliers $e^{2iπθ_1},\ldots,e^{2iπθ_{2d-2}}$. To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of $(\partial \Mand)^{k}$ are dense in the support of the $k^{th}$-bifurcation current $T^k_\bif$ in general families of rational maps, where $\Mand$ is the Mandelbrot set. As a consequence, we also get sharp dimension est…
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 bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an …
We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.
In the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_\bif$ which is called the bifurcation current. This current gives rise to a measure $μ_\bif:=(T_\bif)^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\supp(μ_\bif)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension.
The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $μ_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $μ_\mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $μ_\mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgu…
We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.
In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by H. Zoladek in the article (Eleven small limit cycles in a cubic vector field, Nonlinearity 8, 843--860, 1995) can indeed have eleven limit cycles under perturbations at least …
In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in n-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.
In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\em bifurcation current} as $dd^cL$ and show that it vanishes when the repulsive cycles move holomorphically. We then prove a formula which relates this current with the interaction between the Green current and the current of integration on the critical set. In the 1-dimensional case (i.e. for ${\bf P}^1$) we find a geometrical description of the support of this current and its powers. Finally we introduce the {\em bifurcat…
The dynamics near a border-collision bifurcation are approximated to leading order by a continuous, piecewise-linear map. The purpose of this paper is to consider the higher-order terms that are neglected when forming this approximation. For two-dimensional maps we establish conditions under which a chaotic attractor created in a border-collision bifurcation persists for an open interval of parameters beyond the bifurcation. We apply the results to a prototypical power converter model to prove the model exhibits robust chaos.
We rigorously show that a class of systems of partial differential equations modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). A direct consequence of our result is that the oscillations of the dynamics are \textit{not} due to rotation waves though the system exhibits Euclidean symmetries. The main difficulties of carrying out the program are: 1) the system under study contains multi bifurcation parameters and we do not know \textit{a priori} how they come into play in the bifurcation dynamics. 2) the r…
Let (ρ_λ)_{λ\in Λ} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λdescribing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on the Riemann sphere. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μon G. We…
We study the fine geometric structure of bifurcation currents in the parameter space of cubic polynomials viewed as dynamical systems. In particular we prove that these currents have some laminar structure in a large region of parameter space, reflecting the possibility of quasiconformal deformations. On the other hand, there is a natural bifurcation measure, supported on the closure of rigid parameters. We prove a strong non laminarity statement relative to this measure.
This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only sto…
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$ sufficiently small. Here $g_i:\mathcal{D}\rightarrow\mathbb{R}^n$, for $i=0,1,\ldots,k$, are smooth functions being $\mathcal{D}\subset \mathbb{R}^n$ an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$ x'=F_0(t,x)+\sum_{i=1}^k \vare…
