Equality of pressures for rational functions.
Joint with F. Przytycki and S. Smirnov.
To appear in Ergodic Theory and Dynamical Systems.
Abstract. We prove that for all rational functions $f$ on the Riemann sphere and potential $-t\ln|f'|, t\ge 0$ all the notions of pressure introduced in [P2] coincide. In particular we get a new simple proof of the equality between hyperbolic Hausdorff dimension and minimal exponent of conformal measure on Julia set ([DU] \& [P1]). We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet-Eckmann condition. In Appendix we discuss the case $t<0$.
Bi-analytic elements and partial isometries of hyperbolic space.
Version precedente en francais.
Contemporary Mathematics 319 (2003), 319 - 343.
Abstract.
In this paper we study geometric properties of {\it injective} analytic maps, in the ultrametric setting.
Although this is a classic subject in the complex setting, the ultrametric case has been studied only rather recently.
More precisely we study {\it bi-analytic elements}: injective analytic elements whose inverse is also an analytic element.
Assuming the domain of definition {\it infraconnected} our main result states that any such map can be written as the composition of an isometry and a homography (a map of the form $z \longrightarrow \frac{a z + b}{cz+ d}$, with $ad - bc \neq 0$).
From this result we easily solve a conjecture of A.~Escassut and M.C.~Sarmant.
Our main tool is to consider extensions of analytic elements to (part of) $p$-{\it adic hyperbolic space}.
The main property we prove is that {\it the extension of a bi-analytic element preserves the intrinsic distance of hyperbolic space}.
Une caracterisation des fonctions holomorphes injectives en analyse ultrametrique.
C.R. Acad. Sci. Paris 335 (2002), 441-446.
R\'esum\'e.
On montre qu'une fonction holomorphe non-constante $f$ d\'efinie sur un sous-espace analytique de $\C_p$ est injective si et seulement si on a
$$
\left| \frac{f(x) - f(y)}{x - y} \right|^2 =
|f'(x) \cdot f'(y)|,
\mbox{ pour tous $x$ et $y$ distincts.}
$$
Cette caract\'erisation d\'emontre l'analogue, pour les fonctions holomorphes, d'une conjecture de A.~Escassut et M.C.~Sarmant.
D'autre part on donne une contre-exemple \`a cette conjecture, qui concerne les \'el\'ements bi-analytiques.
Abstract.
We prove that a non constant holomorphic function $f$ defined over an analytic subspace of $\C_p$ is injective if and only if
$$
\left| \frac{f(x) - f(y)}{x - y} \right|^2 =
|f'(x) \cdot f'(y)|,
\mbox{ for every distinct $x$ and $y$.}
$$
This caracterisation proves the analogue, for holomorphic functions, of a conjecture of A.~Escassut and M.C.~Sarmant.
On the other hand we give a counter example to this conjecture, that concerns bi-analytic elements.
Sur la structure des ensembles de Fatou p-adiques.
Weak hyperbolicity on periodic orbits for polynomials.
C.R. Acad. Sci. Paris 334 (2002), 1113-1118.
Abstract. We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + \varepsilon}$, for some $\varepsilon > 0$, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic oribts with ``small'' multipliers. Somehow surprisingly the proof is based in measure theorical considerations.
R\'esum\'e. On d\'emontre que si les multiplicateurs des orbites p\'eriodiques r\'epulsifs d'un polyn\^ome complexe croissent au moins comme $n^{5 + \varepsilon}$, o\`u $\varepsilon > 0$, alors l'ensemble de Julia du polyn\^ome est localement connexe quand il est connexe. Comme cons\'equence on obtient que pour un polyn\^ome complexe l'existence d'un cycle de Cremer implique l'existence d'une suite de orbites p\'eriodiques r\'epulsifs ayant des multiplicateurs ``petites''. D'une fa\c{c}on un puet surprenante la d\'emonstration utilise des arguments de la th\'eorie de la mesure.
L'espace hyperbolique p-adique et dynamique de foncitons rationnelles.
Preprint IMS at Stony Brook 2001/12.
First part, to appear in Compositio Math.,
Second part.
Abstract. We study dynamics of rational maps of degree at least 2 with coefficients in the field $\C_p$, where $p > 1$ is a fixed prime number. The main ingredient is to consider the action of rational maps in $p$-adic hyperbolic space, denoted $\H_p$. Hyperbolic space $\H_p$ is provided with a natural distance, for which it is connected and one dimensional (an $\R$-tree). This advantages with respect to $\C_p$ give new insight into dynamics; in this paper we prove the following results about periodic points. In forthcoming papers we give applications to the Fatou/Julia theory over $\C_p$.
First we prove that the existence of at least two non-repelling periodic points implies de existence of infinitely many of them. This is in contrast with the complex setting where there can be at most finitely many non-repeling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space.
We also caracterise those rational maps with finitely many periodic points in hyperbolic space. Such a rational map can have at most one periodic point (which is then fixed) and we caracterise those rational maps having no periodic points and those rational maps having precisely one periodic point in hyperbolic space.
We also prove a formula relating different objects in the projective line and in hyperbolic space, which are fixed by a given rational map. Finally we relate hyperbolic space in the form given here, to well known objects: the Bruhat-Tits building of $PSL(2, \C_p)$ and the Berkovich space of $\P(\C_p)$.
Equivalence and topological invariance of conditions for non-uniform
hyperbolicity in the iteration of rational maps.
Joint with F. Przytycki and S. Smirnov.
Inventiones Mathematicae 151 (2003), 29-63.
Abstract. We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions (including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components, etc.). The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy.
We show furthermore that for rational maps with one critical point in the Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions; thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement (for the unicritical case) is valid in general.
Rational maps with decay of geometry:
rigidity, Thurston's algorithm and local connectivity.
Preprint IMS at Stony Brook 2000/9.
To appear in Ergodic Theory and Dynamical Systems.
Abstract. We study dynamics of rational maps that satisfy a decay of geometry condition. Well known conditions of non-uniform hyperbolicity, like summability condition with exponent one, imply this condition. We prove that Julia sets have zero Lebesgue measure, when not equal to the whole sphere, and in the polynomial case every connected component of the Julia set is locally connected.
We show how rigidity properties of quasi-conformal maps that are conformal in a big dynamically defined part of the sphere, apply to dynamics. For example we give a partial answer to a problem posed by Milnor about Thurston's algorithm and we give a proof that the Mandelbrot set, and its higher degree analogues, are locally connected at parameters that satisfy the decay of geometry condition. Moreover we prove a theorem about similarities between the Mandelbrot set and Julia sets.
In an appendix we prove a rigidity property that extends a key situation encountered by Yoccoz in his proof of local connectivity of the Mandelbrot set at at most finitely renormalizable parameters.
Dynamique des fonctions rationnelles sur des corps locaux.
The\`ese, Orsay 2000. A appraitre dans Asterisque.
R\'esum\'e. Soit $p > 1$ un nombre premier, $\Q_p$ le corps des nombres $p$-adiques et soit $\C_p$ la plus petite extension compl\`ete et alg\'ebriquement close de $\Q_p$. Ce travail est consacr\'e \`a l'\'etude de la dynamique des fonctions rationnelles sur la droite projective ${\Bbb P}(\C_p)$. A chaque fonction rationnelle $R \in \C_p(z)$ on associe son {\it domaine de quasi-p\'eriodicit\'e}, qui est \'egal \`a l'interieur de l'ensemble des points dans ${\Bbb P}(\C_p)$ qui sont r\'ecurrents par $R$. On donne plusieurs caract\'erisations du domaine de quasi-p\'eriodicit\'e et on d\'ecrit sa dynamique locale et globale. Comme dans le cas complexe on a une partition de la droite ${\Bbb P}(\C_p)$ en l'ensemble de Fatou et l'ensemble de Julia. Par analogie au cas complexe on fait la conjecture de non-errance suivante : tout disque errant est attir\'e par un cycle attractif. On montre que ceci a lieu si et seulement si tout point dans l'ensemble de Fatou est soit attir\'e par un cycle attractif, soit rencontre le domaine de quasi-p\'eriodicit\'e par it\'eration positive. On montre que les composantes du domaine de quasi-p\'eriodicit\'e (qui sont les analogues $p$-adiques des disques des Siegel et des anneaux de Herman) sont des affino\"{\i}des ouverts (c'est-\`a-dire que leur g\'eometrie est simple) et on d\'ecrit la dynamique sur une composante donn\'ee.
Abstract. Let $p > 1$ be a prime number, \ $\Q_p$ the field of $p$-adic numbers and let \ $\C_p$ be the smallest complete extension of \ $\Q_p$ that is algebraically close. This work is dedicated to the study of the dynamics of rational fucntions in the projective line \ ${\Bbb P}(\C_p)$. To each rational function $R \in \C_p(z)$ we associate its {\it quasi-periodicity domain}, which is equal to the interior of the set of points in \ ${\Bbb P}(\C_p)$ that are recurrent by $R$. We give several caracterizations of the quasi-periodicity domain and we describe its local and global dynamics. Like in the complex case there is a partition of the line ${\Bbb P}(\C_p)$ in the Fatou set and the Julia set. By analogy to the complex case we make the following non-wandering conjecture: every wandering disc is attracted to an attracting cycle. We prove that this holds if and only if every point in the Fatou set is either attracted to an attracting cycle or if it is mapped to the quasi-periodicity domain under forward iteration. We prove that analytic components of the domain of quasi-periodicity (which are the $p$-adiqc analogues of Siegel discs and Herman rings) are open affino\"{\i}des (that is, they have simple geometry) and we describe the dynamics in a given component.
On the continuity of Hausdorff dimension of Julia sets
and similarity between the Mandelbrot set and Julia sets.
Fundamenta Mathematicae 170 (2001), 287-317.
Abstract. Given $d \ge 2$ consider the family of polynomials $P_c(z) = z^d + c$, $c \in \C$. Denote by $J_c$ the Julia set of $P_c$ and let ${\cal M}_d = \{ c|\, J_c \mbox{ is connected} \}$ which is a full compact set; for $d = 2$ it is called the Mandelbrot set. We study semihyperbolic parameters $c_0 \in \partial {\cal M}_d$: those for which the critical point $0$ is not recurrent by $P_{c_0}$ and without parabolic cycles. By Shishikura it follows that the Hausdorff dimension of $J_c$, denoted by $HD(J_c)$, is not continuous at such $c_0 \in \partial{\cal M}_d$; on the other hand $c \rightarrow HD(J_c)$ is analytic in $\C - {\cal M}_d$. Our first result asserts that there is still some continuity if one approaches $c_0$ in a good way: there is $C = C(c_0) > 0$ such that if $c_n \rightarrow c_0$ verifies,
$$ dist(c_n, {\cal M}_d) \ge C|c_n - c_0|^{1 + \frac{1}{d}} \, \mbox{ then } \, HD(J_{c_n}) \rightarrow HD(J_{c_0}). $$
A main ingredient is the similarity between ${\cal M}_d$ and $J_{c_0}$ near $c_0$; we prove that the biholomorphism $\psi : \hat{\C} - J_{c_0} \longrightarrow \hat{\C} - {\cal M}_d$ tangent to the identity at infinity is conformal at $c_0$: there is $\lambda \neq 0$ such that,
$$ \psi(z) = c_0 + \lambda(z - c_0) + {\cal O}(|z - c_0|^{1 + \frac{1}{d}}), \, z \not \in J_{c_0}. $$
This easily implies a theorem of Tan Lei with additional estimates. The fact that $\lambda \neq 0$ is related to a transversality phenomenon that is well known for Misiurewicz parameters and we extend to the semihyperbolic case. We also prove that if $d_H$ denotes the Hausdorff distance then,
$$ d_H(J_c, J_{c_0}), d_H(K_c, J_{c_0}) = {\cal O}(|c - c_0|^{\frac{1}{d}}, $$
On the topology of arithmetic sums of regular Cantor sets.
Joint with C. Moreira and E. Munoz.
Nonlinearity 13 (2000), no. 6, 2077-2087.
Rigid annuli, Thurston's pull-back argument and the
Collet Eckmann condition.