125 research outputs found
The supports of higher bifurcation currents
Let (f_\lambda) 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 \Lambda. To this data is associated a closed positive
current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, 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).Comment: 13 page
Bifurcation currents and equidistribution on parameter space
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.Comment: Revised version, 46 pages, to appear in the proceedings of the
conference "Frontiers in complex dynamics (Celebrating John Milnor's 80th
birthday)
Wermer examples and currents
In this paper we give the first examples of positive closed currents in
with continuous potentials, vanishing self-intersection, and
which are not laminar. More precisely, they are supported on sets "without
analytic structure". The result is mostly interesting when the potential has
regularity close to , because laminarity is expected to hold in that case.
We actually construct examples which are for all .Comment: Minor modifications. Final version, to appear in GAF
Fatou directions along the Julia set for endomorphisms of CP^k
Not much is known about the dynamics outside the support of the maximal
entropy measure for holomorphic endomorphisms of . In this
article we study the structure of the dynamics on the Julia set, which is
typically larger than . The Julia set is the support of the
so-called Green current , so it admits a natural filtration by the supports
of the exterior powers of . For , let . We
show that for a generic point of there are at least
"Fatou directions" in the tangent space. We also give estimates for the
rate of expansion in directions transverse to the Fatou directions.Comment: Final, shorter version, to appear in J. Math. Pures App
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