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Geometry of currents, intersection theory and dynamics of horizontal-like maps

Tien-Cuong Dinh, Nessim Sibony (2006)

Annales de l’institut Fourier

We introduce a geometry on the cone of positive closed currents of bidegree ( p , p ) and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.

Hubbard trees

Alfredo Poirier (2010)

Fundamenta Mathematicae

We provide a full classification of postcritically finite polynomials as dynamical systems by means of Hubbard trees. The information encoded in these objects is solid enough to allow us recover all the relevant statical and dynamical aspects of the corresponding Julia sets.

Hyperbolic components of the complex exponential family

Robert L. Devaney, Nuria Fagella, Xavier Jarque (2002)

Fundamenta Mathematicae

We describe the structure of the hyperbolic components of the parameter plane of the complex exponential family, as started in [1]. More precisely, we label each component with a parameter plane kneading sequence, and we prove the existence of a hyperbolic component for any given such sequence. We also compare these sequences with the more commonly used dynamical kneading sequences.

Immediate and Virtual Basins of Newton’s Method for Entire Functions

Sebastian Mayer, Dierk Schleicher (2006)

Annales de l’institut Fourier

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.

Intertwined internal rays in Julia sets of rational maps

Robert L. Devaney (2009)

Fundamenta Mathematicae

We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.

Introduction

Pascale Roesch (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

La mesure d’équilibre d’un endomorphisme de ℙ k ( ℂ )

Xavier Buff (2004/2005)

Séminaire Bourbaki

Soit f un endomorphisme holomorphe de ℙ k ( ℂ ) . Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité μ ayant les propriétés suivantes : μ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de f s’équidistribuent par rapport à μ et μ est l’unique mesure d’entropie maximale de f .

Linear Fractional Recurrences: Periodicities and Integrability

Eric Bedford, Kyounghee Kim (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Linear fractional recurrences are given as z n + k = A ( z ) / B ( z ) , where A ( z ) and B ( z ) are linear functions of z n , z n + 1 , ⋯ , z n + k - 1 . In this article we consider two questions about these recurrences: (1) Find A ( z ) and B ( z ) such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each k there are k -step linear fractional recurrences...

Local dynamics of holomorphic diffeomorphisms

Filippo Bracci (2004)

Bollettino dell'Unione Matematica Italiana

This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.

Markov partitions for fibre expanding systems

Manfred Denker, Hajo Holzmann (2008)

Colloquium Mathematicae

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

Matings and the other side of the dictionary

John Hubbard (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

In the theory of rational maps an important role is played by matings. These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of double limits, that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and...

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