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Dynamics of quadratic polynomials : complex bounds for real maps

Mikhail Lyubich, Michael Yampolsky (1997)

Annales de l'institut Fourier

We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z z 2 + c , c [ - 2 , 1 / 4 ] , is locally connected.

Hausdorff dimension and measures on Julia sets of some meromorphic maps

Krzysztof Barański (1995)

Fundamenta Mathematicae

We study the Julia sets for some periodic meromorphic maps, namely the maps of the form f ( z ) = h ( e x p 2 π i T z ) where h is a rational function or, equivalently, the maps ˜ f ( z ) = e x p ( 2 π i h ( z ) ) . When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1....

Hyperbolic measure of maximal entropy for generic rational maps of k

Gabriel Vigny (2014)

Annales de l’institut Fourier

Let f be a dominant rational map of k such that there exists s < k with λ s ( f ) > λ l ( f ) for all l . Under mild hypotheses, we show that, for A outside a pluripolar set of Aut ( k ) , the map f A admits a hyperbolic measure of maximal entropy log λ s ( f ) with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of f to k + 1 . This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic...

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...

Normal forms of invariant vector fields under a finite group action.

Federico Sánchez-Bringas (1993)

Publicacions Matemàtiques

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

On backward stability of holomorphic dynamical systems

Genadi. Levin (1998)

Fundamenta Mathematicae

For a polynomial with one critical point (maybe multiple), which does not have attracting or neutral periodic orbits, we prove that the backward dynamics is stable provided the Julia set is locally connected. The latter is proved to be equivalent to the non-existence of a wandering continuum in the Julia set or to the shrinking of Yoccoz puzzle-pieces to points.

On complexification and iteration of quasiregular polynomials which have algebraic degree two

Ewa Ligocka (2005)

Fundamenta Mathematicae

We prove that each degree two quasiregular polynomial is conjugate to Q(z) = z² - (p+q)|z|² + pqz̅² + c, |p| < 1, |q| < 1. We also show that the complexification of Q can be extended to a polynomial endomorphism of ℂℙ² which acts as a Blaschke product (z-p)/(1-p̅z) · (z-q)/(1-q̅z) on ℂℙ²∖ℂ². Using this fact we study the dynamics of Q under iteration.

On the C⁰-closing lemma

Anna A. Kwiecińska (1996)

Annales Polonici Mathematici

A proof of the C⁰-closing lemma for noninvertible discrete dynamical systems and its extension to the noncompact case are presented.

Points périodiques d’applications birationnelles de 2

Charles Favre (1998)

Annales de l'institut Fourier

Nous donnons une condition suffisante pour l’existence de points périodiques pour une application birationnelle de 2 . Sous cette hypothèse, une estimation précise du nombre de points périodiques de période fixée est obtenue. Nous donnons une application de ce résultat à l’étude dynamique de ces applications, en calculant explicitement l’auto-intersection de leur courant invariant naturellement associé. Nos résultats reposent essentiellement sur le théorème de Bézout donnant le cardinal de l’intersection...

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