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Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle

Arnaud Chéritat (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We...

The Bernoulli shift as a basic chaotic dynamical system

Kučera, Václav (2019)

Programs and Algorithms of Numerical Mathematics

We give a brief introduction to the Bernoulli shift map as a basic chaotic dynamical system. We give several examples where the iterates of a~mapping can be understood using the Bernoulli shift. Namely, the iteration of real interval maps and iteration of quadratic functions in the complex plain.

The characteristic variety of a generic foliation

Jorge Vitório Pereira (2012)

Journal of the European Mathematical Society

We confirm a conjecture of Bernstein–Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.

The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let M be a two-dimensional complex manifold and f : M → M a holomorphic map. Let S ⊂ M be a curve made of fixed points of f , i.e.  Fix ( f ) = S . We study the dynamics near  S in case  f acts as the identity on the normal bundle of the regular part of  S . Besides results of local nature, we prove that if  S is a globally and locally irreducible compact curve such that S · S < 0 then there exists a point p ∈ S and a holomorphic f -invariant curve with  p on the boundary which is attracted by  p under the action of  f . These results are achieved...

The dynamics of two-circle and three-circle inversion

Daniel M. Look (2008)

Fundamenta Mathematicae

We study the dynamics of a map generated via geometric circle inversion. In particular, we define multiple circle inversion and investigate the dynamics of such maps and their corresponding Julia sets.

The supports of higher bifurcation currents

Romain Dujardin (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on ℙ 1 ( ℂ ) , with k marked critical points c 1 , ... , c k . To this data is associated a closed positive current T 1 ∧ ⋯ ∧ 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 ∧ ⋯ ∧ T k ) .

Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics

Mariusz Urbański (2003)

Fundamenta Mathematicae

We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately...

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