In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincare exponent for the Fibonacci map is less than two, which implies that the Hausdor ff dimension of its Julia set is less than two.

In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincare exponent for the Fibonacci map is less than two, which implies that the Hausdor ff dimension of its Julia set is less than two.

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. In this thesis, we review existing results on rational maps sharing a Julia set, and offer results of our own on the symmetry group of such maps. / Desde a década de oitenta, um enorme progresso foi feito no problema de determinar quais funções têm o mesmo conjunto de Julia. O caso polinomial foi completamente respondido em 1995, e mostrou-se que as simetrias do conjunto de Julia têm um papel central nessa questão. O caso racional permanece aberto, mas já se sabe que ele é muito mais complexo do que o polinomial. Nesta dissertação, nós revisamos resultados existentes sobre aplicações racionais com o mesmo conjunto de Julia e apresentamos nossos próprios resultados sobre o grupo de simetrias de tais aplicações.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

京都大学 / 新制・課程博士 / 博士(人間・環境学) / 甲第23273号 / 人博第988号 / 新制||人||234(附属図書館) / 2020||人博||988(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

Dans cette thèse, on s'intéresse aux systèmes dynamiques holomorphes dépendants de paramètres. Notre objectif est de contribuer à une théorie de la stabilité et des bifurcations en plusieurs variables complexes, généralisant celle des applications rationnelles fondées sur les travaux de Mané, Sad, Sullivan et Lyubich. Pour une famille d'applications d'allure polynomiale, on prouve l'équivalence de plusieurs notions de stabilité, entre autres une version asymptotique du mouvement holomorphe des cycles répulsifs et d'un sous-ensemble de l'ensemble de Julia de mesure pleine. Cela peut etre considéré comme une généralisation mesurable à plusieurs variables du célèbre lambda-lemme et nous permet de dégager un concept cohérent de stabilité dans ce cadre. Après avoir compris les bifurcations holomorphes, on s'intéresse à la continuité Hausdorff des ensembles de Julia. Nous relions cette propriété à l'existence de disques de Siegel dans l'ensemble de Julia, et donnons un exemple de ce phénomène. Finalement, on étudie la continuité du point de vue de l'implosion parabolique. Nous établissons un théorème de Lavaurs deux-dimensionel, ce qui nous permet d'étudier des phénomènes de discontinuité pour des perturbations d'applications tangentes à l'identité. / In this thesis we study holomorphic dynamical systems depending on parameters. Our main goal is to contribute to the establishment of a theory of stability and bifurcation in several complex variables, generalizing the one for rational maps based on the seminal works of Mané, Sad, Sullivan and Lyubich. For a family of polynomial like maps, we prove the equivalence of several notions of stability, among the others an asymptotic version of the holomorphic motion of the repelling cycles and of a full-measure subset of the Julia set. This can be seen as a measurable several variables generalization of the celebrated lambda-lemma and allows us to give a coherent definition of stability in this setting. Once holomorphic bifurcations are understood, we turn our attention to the Hausdorff continuity of Julia sets. We relate this property to the existence of Siegel discs in the Julia set, and give an example of such phenomenon. Finally, we approach the continuity from the point of view of parabolic implosion and we prove a two-dimensional Lavaurs Theorem, which allows us to study discontinuities for perturbations of maps tangent to the identity.

Cette thèse est consacrée à l'étude de quelques problèmes en dynamique holomorphe discrete et continue à l'aide de la Théorie du Pluripotentiel. Le premier problème présenté concerne la description des paires d'endomorphismes holomorphes permutables du plan projectif complexe qui ne partagent pas une itérée. Nous nous intéressons au cas où les degrés des deux applications coïncident après un certain nombre d'itérations. Nous montrons que telles applications sont des exemples de Lattès ou bien des relèvements des exemples de Lattès unidimensionnels. Combiné avec un théorème de T.-C. Dinh et N. Sibony ce résultat complète la classification des paires permutables en dimension deux. Ensuite, nous nous intéressons à la dynamique des laminations par variétés complexes. Nous montrons que, dans une variété kählérienne compacte, le carré de la classe de cohomologie d'un cycle feuilleté dirigé par une lamination transversalement Lipschitz est toujours zéro. Parmi les conséquences nous montrons que l'espace projectif complexe $\pr^{n}$ n'admet pas de cycle feuilleté transversalement Lipschitz de dimension $q \leq \frac{n}{2}$. Cela généralise un résultat de J.E. Forn\ae ss et N. Sibony. Dans la dernière partie nous étudions les mesures de Monge-Ampère à potentiel höldérien. Nous montrons que ces mesures satisfont un analogue d'un théorème de H. Skoda concernant l'intégrabilité exponentielle d'une fonction plurisousharmonique en termes de ses nombres de Lelong. Ce résultat peut être vu comme une très forte compacité pour les fonctions plurisousharmoniques qui sont eux-mêmes un outil fondamental en dynamique holomorphe. / This thesis is devoted to the study of some problems in discrete and continuous holomorphic dynamics with the tools of Pluripotential Theory. The first problem we consider involves the description of commuting pairs of holomorphic endomorphisms of the complex projective plane that do not share an iterate. We consider the case when their degrees coincide after some number of iterations. We show that these maps are either Lattès maps or lifts of one-dimensional Lattès maps. Together with a theorem of T.-C. Dinh and N. Sibony this result completes the classification of commuting pairs in dimension two. Later on, we turn our attention to the dynamics of laminations by complex manifolds. We show that, on a compact Kähler manifold, the square of the cohomology class of a foliated cycle directed by a transversally Lipschitz lamination is always zero. As a corollary we show that the complex projective space $\pr^n$ do not carry any transversally Lipschitz foliated cycle of dimension $q \leq \frac{n}{2}$, generalizing a result by J.E. Forn\ae ss and N. Sibony. In the last part we study Monge-Ampère measures with Hölder continuous potential. We show that these measures satisfy an analogue of a theorem of H. Skoda concerning the exponential integrability of plurisubharmonic functions in terms of its Lelong numbers. This result can be viewed as a strong compactness property of plurisubharmonic functions, a class of functions of fundamental importance in holomorphic dynamics.