Indiana University-Purdue University Indianapolis (IUPUI) / Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.

[pt] Neste texto estudamos diversos aspectos de singularidades de campos vetoriais holomorfos em dimensão 2. Discutimos detalhadamente o caso particular de uma singularidade sela-nó e o papel desempenhado pelas normalizações setoriais. Isto nos conduz à classificação analítica de difeomorfismos tangentes à identidade. seguir abordamos o Teorema de Seidenberg, tratando da redução de singularidades degeneradas em singularidades simples, através do procedimento de blow-up. Por fim, estudamos a demonstração do Teorema de Mattei-Moussu, acerca da existência de integrais primeiras para folheações holomorfas. / [en] In the present text, we study the different aspects of singularities of holomorphic vector fields in dimension 2. We discuss in detail the particular case of a saddle-node singularity and the role of the sectorial normalizations. This leads us to the analytic classiffication of diffeomorphisms which are tangent to the identity. Next, we approach the Seidenberg Theorem, dealing with the reduction of degenerated singularities into simple ones, by means of the blow-up procedure. Finally, we study the proof of the well-known Mattei-Moussu Theorem concerning the existence of first integrals to holomorphic foliations.

La thèse traite de l’utilisation des systèmes hybrides dans le contexte particulier des communications sécurisées et de la cryptographie. Ce travail est motivé par les faits suivants. L’essor considérable des communications qui a marqué ces dernières décennies nécessite des besoins croissants en terme de sécurité des échanges et de protection de l’information. Dans ce contexte, la cryptographie joue un rôle central puisque les informations transitent la plupart du temps au travers de canaux publics. Parmi les nombreuses techniques de chiffrement existants, le chiffrement par flot se distingue tout particulièrement lorsqu’on le débit d’une communication sécurisée est privilégié. Les chiffreurs par flot sont construits à partir de générateurs de séquences complexes décrits par des systèmes dynamiques et devant être synchronisés de part et d’autre du canal d’échanges. Les objectifs et les résultats de ce travail se déclinent en trois points. Tout d’abord, l’intérêt d’utiliser des systèmes hybrides en tant que primitives cryptographiques est motivé. Par la suite, une étude comparative est menée afin d’établir une connexion entre les algorithmes de masquage de l’information basés sur le chaos et les algorithmes de chiffrement usuels. L’étude porte exclusivement sur des considérations structurelles et repose sur des concepts de la théorie du contrôle, en particulier l’inversibilité à gauche et la platitude. On montre que la technique de masquage dite par inclusion, qui consiste à injecter l’information à protéger dans une dynamique complexe, est la plus efficace. De plus, on montre que sous la condition de platitude, un système de masquage par inclusion est structurellement équivalent à un chiffreur par flot particulier appelé auto-synchronisant. Enfin, des méthodes de cryptanalyse pour évaluer la sécurité du masquage par inclusion sont proposées pour une classe particulières de systèmes hybrides à savoir les systèmes linéaires à commutations. A nouveau, des concepts de la théorie du contrôle sont utilisés, il s’agit de l’identifiabilité paramétrique et des algorithmes d’identification. Des spécificités relatives au contexte particulier de la cryptographie sont prises en compte. En effet, contrairement à la plupart des cas rencontrés dans le domaine du contrôle où les variables des modèles dynamiques sont continues car relatives à des systèmes physiques, les variables prennent ici des valeurs discrètes. Les modèles dynamiques sont en effet décrits non plus dans le corps des réels mais dans des corps finis en vue d’une implémentation sur des machines à états finis tels ordinateur ou tout autre dispositif numérique / This manuscript deals with a specific engineering application involving hybrid dynamical systems : secure communications and cryptography. The work is motivated by the following facts. The considerable progress in communication technology during the last decades has led to an increasing need for security in information exchanges. In this context, cryptography plays a major role as information is mostly conveyed through public networks. Among a wide variety of cryptographic techniques, stream ciphers are of special interest for high speed encryption. They are mainly based on generators of complex sequences in the form of dynamical systems, which must be synchronized at the transmitter and receiver sides. The aim of this work is threefold. First, the interest of resorting to hybrid dynamical systems for the design of cryptographic primitives is motivated. Secondly, a connection between chaotic and conventional cryptography is brought out by comparing the respective algorithms proposed in the open literature. The investigation focuses on structural consideration. Control theoretical concepts, in particular left invertibility and flatness, are the central tools to this end. It is shown that the so-called message-embedding technique, consisting in injecting the information to be concealed into a dynamical system, is the most relevant technique. Furthermore, it is shown that, under the flatness condition, the resulting cipher acts as a self-synchronizing stream cipher. Finally, cryptanalytic methodologies for assessing the security of the message-embedded cryptosystem involving a special class of hybrid systems, namely the switched linear systems, are proposed. Again concepts borrowed from control theory, namely identifiability and identification, are considered. Specificities related to the context are taken into account. The variables describing the dynamical systems do not take values in a continuum unlike what usually happens in automatic control when physical models are considered. They rather take values in finite cardinality sets, especially finite fields, since an implementation in finite state machines, say computers or digital electronic devices, is expected

In this master thesis we investigate, from a topological point of view and without applying Thurston´s Theorem, why the mating of the so called basilica polynomial <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D(z)=z%5E%7B2%7D-1" /> and the dendrite <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7Bi%7D(z)=z%5E%7B2%7D+i" /> is shared with the mating of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> and the dendrite <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-i%7D(z)=z%5E%7B2%7D-i" />. Both these matings equal the rational map <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D(z)=%5Cfrac%7B3%7D%7Bz%5E%7B2%7D+2z%7D" />. Defined in the thesis are for both matings homeomorphic changes of coordinates<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpsi_%7B-1%7D%5E%7B%5Cpm%7D" /> from the set <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L=%5Coverset%7B%5Ccirc%7D%7BK%7D%5Cleft(f_%7B-1%7D%20%5Cright)%5Ccup%5Cleft(%5Ccup_%7Bn=0%7D%5E%7B%5Cinfty%7Df_%7B-1%7D%5E%7B%5Ccirc(-n)%7D(z_%7B%5Calpha%7D)%5Cright)" /> to the Fatou and Julia set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D" />. Here <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" /> is the filled Julia set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z_%7B%5Calpha%7D" /> is the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />-fixed point of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" />. / I detta examensarbete undersöker vi, från en topologisk synvinkel och utan applicering av Thurstons teorem, varför matchningen av det så kallade basilikapolynomet <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D(z)=z%5E%7B2%7D-1" /> och dendriten <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7Bi%7D(z)=z%5E%7B2%7D+i" /> är delad med matchningen av <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> och dendriten <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-i%7D(z)=z%5E%7B2%7D-i" />. Båda dessa matchningar är lika med den rationella avbildningen  <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D(z)=%5Cfrac%7B3%7D%7Bz%5E%7B2%7D+2z%7D" />. Definierat i examensarbetet är för båda matchningarna homoemorfa koordinatbyten<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpsi_%7B-1%7D%5E%7B%5Cpm%7D" /> från mängden<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L=%5Coverset%7B%5Ccirc%7D%7BK%7D%5Cleft(f_%7B-1%7D%20%5Cright)%5Ccup%5Cleft(%5Ccup_%7Bn=0%7D%5E%7B%5Cinfty%7Df_%7B-1%7D%5E%7B%5Ccirc(-n)%7D(z_%7B%5Calpha%7D)%5Cright)" /> till Fatou- och Juliamängden av <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D" />. Här är <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" /> den ifyllda Juliamängden av avbildningen <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> och <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z_%7B%5Calpha%7D" /> är den <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />-fixerade punkten i <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" />.

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

京都大学 / 0048 / 新制・論文博士 / 博士(理学) / 乙第13201号 / 論理博第1560号 / 新制||理||1635(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 宍倉 光広, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第2項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(人間・環境学) / 甲第24698号 / 人博第1071号 / 新制||人||251(附属図書館) / 2022||人博||1071(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)准教授 木坂 正史, 教授 角 大輝, 教授 足立 匡義, 教授 諸澤 俊介 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. / Generalizamos as noções de estabilidade estrutural e hiperbolicidade para a família de correspondências holomorfas Hc(z) = zr + c; onde r > 1 é racional e zr = exp r log z: Descobrimos que Hc é estruturalmente estável em todos os parâmetros hiperbólicos satisfazendo a condição de fuga. Tipicamente Hc possui infinitos pontos periódicos atratores, fato totalmente inesperado, uma vez que este número é sempre finito para aplicações racionais. O conjunto de tais pontos dá origem ao chamado conjunto de Julia dual, que é um conjunto de Cantor proveniente de um Conformal Iterated Function System. Tanto o conjunto de Julia e quanto seu dual são projeções de movimentos holomorfos de sistemas definidos em subconjuntos compactos denotados por Xc e Wc; respectivamente de um espaço de Banach. Para todo c próximo de zero: (1) mostramos que Jc é reunião de arcos quase-conformes próximos do círculo unitário; (2) o conjunto Xc é um movimento holomorfo do solenóide X0; (3) utilizando o formalismo dos estados de Gibbs, exibimos um limitante superior para a dimensão de Hausdorff de Jc. Consequentemente, Jc possui medida de Lebesgue nula.

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. / Generalizamos as noções de estabilidade estrutural e hiperbolicidade para a família de correspondências holomorfas Hc(z) = zr + c; onde r > 1 é racional e zr = exp r log z: Descobrimos que Hc é estruturalmente estável em todos os parâmetros hiperbólicos satisfazendo a condição de fuga. Tipicamente Hc possui infinitos pontos periódicos atratores, fato totalmente inesperado, uma vez que este número é sempre finito para aplicações racionais. O conjunto de tais pontos dá origem ao chamado conjunto de Julia dual, que é um conjunto de Cantor proveniente de um Conformal Iterated Function System. Tanto o conjunto de Julia e quanto seu dual são projeções de movimentos holomorfos de sistemas definidos em subconjuntos compactos denotados por Xc e Wc; respectivamente de um espaço de Banach. Para todo c próximo de zero: (1) mostramos que Jc é reunião de arcos quase-conformes próximos do círculo unitário; (2) o conjunto Xc é um movimento holomorfo do solenóide X0; (3) utilizando o formalismo dos estados de Gibbs, exibimos um limitante superior para a dimensão de Hausdorff de Jc. Consequentemente, Jc possui medida de Lebesgue nula.

Associated to any finite simple graph Γ is the <i>chromatic polynomial </i>PΓ(q) whose complex zeros are called the <i>chromatic zeros </i>of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γ_n}∞_<i>n</i>-0 built recursively using a substitution rule expressed in terms of a generating graph. For each <i>n</i>, let <i>μn</i> denote the probability measure that assigns a Dirac measure to each chromatic zero of Γ_<i>n</i>. Under a mild hypothesis on the generating graph, we prove that the sequence <i>μn</i> converges to some measure <i>μ</i> as <i>n</i> tends to infinity. We call <i>μ</i> the limiting measure of <i>chromatic zeros</i> associated to {Γ_n}∞_<i>n-</i>0. In the case of the Diamond Hierarchical Lattice we prove that the support of <i>μ</i> has Hausdorff dimension two.<div><br></div><div>The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove anew equidistribution theorem that can be used to relate the chromatic zeros of ahierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.<br></div>