arXiv subjects

Romain Dujardin

Publications and source records attributed to Romain Dujardin.

46 records · Page 3Linked to original sources

arXiv2009

Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection.

math.CV↗

arXiv2007

Distribution of rational maps with a preperiodic critical point

Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) is periodic converge as a sequence of positive closed (1,1) currents to the bifurcation current attached to c and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree d. By intersecting the d-1 currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure of finite mass which is supported on the connectedness locus. They showed that its support is included in the closure of the set of parameters admitting d-1 neutral cycles. We show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials (i.e. of Misiurewicz points).

math.DS↗

arXiv2006

Continuity of Lyapunov exponents for polynomial automorphisms of $\mathbb{C}^2$

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a 1-dimensional map.

math.DS↗

arXiv2006

Cubic polynomials: a measurable view on parameter space

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.

math.DS↗

arXiv2005

Laminar currents and birational dynamics

We study the dynamics of a bimeromorphic selfmap of a compact complex Kähler surface $X$. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of $\mathbb{C}^2$. This extends recent results by E. Bedford and J. Diller.

math.DS↗

arXiv2005

Structure properties of laminar currents on $\mathbb{P}^2$

We study the structure of a class of laminar closed positive currents on $\mathbb{CP}^2$, naturally appearing in birational dynamics. We prove such a current admits natural non intersecting {\em leaves}, that are closed under analytic continuation. As a consequence it can be seen as a {\em foliation cycle} a weak lamination.

math.CV↗

arXiv2005

Approximation des fonctions lisses sur certaines laminations

We show that on a Riemann surface lamination locally embedded in $\mathbb{C}^2$, $C^1$ functions (in the sense of the $C^1$ structure of the lamination) are uniform limits of ambient $C^1$ functions, with $L^p$ control on the derivatives along the leaves. This implies that locally in $C^2$, a (1,1) positive closed current dominated by a uniformly laminar current is itself uniformly laminar.

math.CV↗

arXiv2004

Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the connectivity of the Julia set, that carries over for certain H{é}non-like and birational maps. We study the structure of disconnected Julia sets and the associated invariant currents. This provides a simple approach to some results of Bedford-Smillie, as well as some new corollaries --the connectedness locus is closed, construction of external rays in the general case, etc. We also prove the following theorem: a hyperbolic polynomial diffeomorphism of C^2 with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.

math.DS↗

arXiv2004

On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$

We construct the Green current for a random iteration of "horizontal-like" mappings in two complex dimensions. This is applied to the study of a polynomial map $f:\mathbb{C}^2\to\mathbb{C}^2$ with the following properties: 1. infinity is $f$-attracting, 2. $f$ contracts the line at infinity to a point not in the indeterminacy set. Then the Green current of $f$ can be decomposed into pieces associated with an itinerary determined by the indeterminacy points. We also study the set of escape rates near infinity, i.e. the possible values of the function $\limsup \frac{1}{n}\log^+\log^+ \norm{f^n}$. We exhibit examples for which this set contains an interval.

math.DS↗

arXiv2003

Sur l'intersection des courants laminaires

We try to find a geometric interpretation of the wedge product of positive closed laminar currents in $\mathbb{C}^2$. We say such a wedge product is geometric if it is given by intersecting the disks filling up the currents. Uniformly laminar currents do always intersect geometrically in this sense. We also introduce a class of ``strongly approximable'' laminar currents, natural from the dynamical point of view, and prove that such currents intersect geometrically provided they have continuous potentials.

math.CV↗