Thesis advisor: Kathryn Lindsey / We use interval maps to construct pseudo-Anosovs and relate important invariants of each regime. This work builds on techniques of André de Carvalho, Toby Hall, Bill Thurston, and others. We introduce a new perspective on the pseudo-Anosovs created in this way, showing how they constitute the vertices of a tree whose edges encode relations between them. We also characterize the pseudo-Anosovs arising from interval maps, and use this result to reprove a universal lower bound on their stretch factors originally due to Boissy-Lanneau. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Les strates de l'espace de modules des di__erentielles ab_eliennes sont naturellementmunies d'un feuilletage holomorphe, appel_e feuilletage isop_eriodique (ou feuilletagesdes p_eriodes aboslues, ou encore feuilletage du noyau). Celui-ci a _et_e introduit il y a25 ans, d'abord par A. Eskin et M. Kontsevitch, puis par K. Calta et C. McMullenavant de devenir un objet important en dynamique de Teichmuller. La questiong_en_erale abord_ee dans ce texte est la suivante :Comment les feuilles du feuilletage isop_eriodique se r_epartissent-ellesdans l'espace de module ?McMullen a d_emontr_e l'ergodicit_e du feuilletage dans les strates principales (o_u toutesles singularit_es sont simples) en genre 2 et 3 en utilisant des techniques issue dela dynamique homog_ene. Calsamiglia, Deroin & Francaviglia ont ensuite _etenduce resulat et obtenu une classi_cation _a la Ratner des ensembles ferm_es satur_espar le feuilletage. Simultan_ement, Hamenstadt a fourni une preuve alternative del'ergodicit_e, toujours dans la strate principale. De fa_con _etonnante, le seul r_esulatconnu pour les autres strates est d^u _a P. Hooper et B. Weiss : les feuilles des surfacesde Arnoux-Yoccoz sont denses dans les strates qui les contiennent.La question de la dynamique du feuilletage isop_eriodique peut ^etre formul_ee dansle contexte plus g_en_eral des sous vari_et_es a_nes. Avila, Eskin et Moller ont prouv_eque la codimension des feuilles est alors paire. Le cas de la codimension 2, ou rang1, est d_ej_a riche. Nous _etablissons un cri_ete de densit_e des feuilles et l'appliquons_a di__erentes familles de vari_et_es a_nes de rang 1. Parmi celles-la, les lieux Prymoccupent une place importante. Nous d_emontrons dans ce cadre que les feuilles sontsoit ferm_ees, soit denses, en fonction de l'artithm_eticit_e du lieu. Dans le cas nonarithm_etique, nous prouvons que le feuilletage est ergodique pour la mesure a_neassoci_ee. Cela aboutit _a la d_ecouverte de nouvelles feuilles denses dans des strates _asingularit_es multiples. Ces r_esultats sugg_erent une connection entre la g_eometrie desvari_et_es a_nes et la dynamique isop_eriodique. L'exploitation de cette connection engenre 3 aboutit _a la classi_cation des vari_et_es a_nes non arithm_etiques ne provenantpas d'orbites ferm_ees dans les strates _a deux singularit_es. / The strata of the moduli space of abelian di_erentials are endowed with a naturalholomorphic foliation, known as the isoperiodic foliation (or absolute period foliationor kernel foliation). It has been introduced 25 years ago by A. Eskin and M. Kontsevichand later by K. Calta and C. McMullen before it became a central object inTeichmuller dynamics. The general question addressed in this text is the following:How do the leaves of the isoperiodic foliation wander around in themoduli space ?McMullen proved the ergodicity of the foliation in the principal stratum (where thesingularities of the abelian di_erentials are all simple) in genus 2 and 3 using resultsfrom group actions on homogeneous space. Calsamiglia, Deroin & Francavigliageneralized this result in higher genera and obtained a Ratner-like classi_cation ofthe closed saturated subsets. Simultaneously, Hamenstadt gave an alternative proofof the ergodicity. Surprisingly enough, for the strata where at least one zero isnot simple, the only result available was due to Hooper and Weiss: the leaf of theArnoux-Yoccoz surface is dense in the stratum in which it belongs.The question of the dynamics of the isoperiodic foliation can be rephrased in the moregeneral context of a_ne manifolds. Avila, Eskin, M^oller proved that the codimensionof the leaves is even. The codimension 2 case, also known as rank 1, already displaysa rich and contrasted picture. We give a criterion for density of the leaves, and applyit to di_erent families of rank one a_ne manifolds. Among those, special attention isdedicated to the Prym eigenform loci. We prove that the leaves are either compactor dense, depending on the arithmeticity of the locus. In the non arithmetic case, weprove that the foliation is ergodic with respect to the a_ne measure. In turn, thisgives new examples of dense leaves in strata where at least one of the singularity isnot simple. The aforementioned results suggest a connection between the dynamicsof the isoperiodic foliation and the geometry of a_ne manifolds. This connection isanalyzed in genus 3 and results in a classi_cation of the proper non arithmetic a_nemanifolds in strata with 2 singularities.

[pt] Automorfismos genéricos de cubos com alças (handlebodies) aparecem do estudo de classes the isotopia de automorfismos de variedades orientáveis de dimensão três. Automorfismos genéricos permanecem como uma das partes menos entendidas desse estudo.Dado um automorfismo genérico de um cubo com alças, é conhecida uma forma de se construir uma laminação bidimensional que é invariante pelo automorfismo. A essa laminação se associa um fator de crescimento. É sabido que, no caso de tal fator de crescimento ser minimal - uma característica importante, pois mede a complexidade essencial do automorfismo - a laminação deve gozar de uma certa propriedade de incompressibilidade. Nessa tese mostramos que o processo de se achar uma laminação com tal propriedade é algoritmico. Por outro lado, mostramos que tal propriedade não garante que o respectivo fator de crescimento seja minimal. Propomos uma outra propriedade, tensão transversal, mais forte que incompressibilidade, que conjecturamos também ser condição necessária para que o fator de crescimento seja minimal. Provamos a conjectura em alguns casos.Além dos resultados mencionados acima, desenvolvemos métodos para gerar automorfismos genéricos de cubos com alcas, que usamos para apresentar alguma variedade de exemplos. / [en] Generic automorphisms of handlebodies appear naturally in the study of isotopy classes of automophisms of orientable three-dimensional manifolds. Generic automorphisms remain as one of the least understood parts of this study. Given a generic automorphism of a handlebody one can construct a bidimensional lamination that is invariant under the automorphism. There is a growth rate associated to this lamination. It is known that, when this growth rate is minimal among all possible choices (an important property, for it measures the essential complexity of the automorphism), the lamination must have a certain incompressibility property. On this thesis we show that the process of finding a lamination with such a property is algorithmic. On the other hand, we show that this said incompressibility property is not sufficient for the minimality of the growth rate. We propose a stronger property, which we called transverse tightness, and conjecture that it is a necessary condition for the growth rate to be minimal. We prove the conjecture in some particular cases. In addition to the results mentioned above, we develop methods to generate generic automorphisms of handlebodies, which we use to present some variety of examples.

Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number \lambda(f). The number \lambda(f) is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur. Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on Sg with orientable foliations whose stretch factor \lambda has algebraic degree 2g. Moreover, the stretch factor \lambda is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d≤g. Our examples also give a new approach to a conjecture of Penner.

Thesis advisor: John A. Baldwin / We classify genus-two L-space knots in S3 and the Poincare homology sphere.This leads to the first and to-date only detection results in knot Floer homology for knots of genus greater than one. Our proofs interweave Floer-homological properties of L-space knots, the geometry of pseudo-Anosov maps, and the theory of train tracks and folding automata for braids. The crux of our argument is a complete classification of fixed-point-free pseudo-Anosov maps in all but one stratum on the genus-two surface with one boundary component. To facilitate our classification, we exhibit a small family of train tracks carrying all pseudo-Anosov maps in most strata on the marked disk. As a consequence of our proof technique, we almost completely classify genus-two, hyperbolic, fibered knots with knot Floer homology of rank 1 in their next-to-top grading in any 3-manifold. Several corollaries follow, regarding the Floer homology of cyclic branched covers, SU(2)-abelian Dehn surgeries, Khovanov and annular Khovanov homology, and instanton Floer homology. / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Dans le premier chapitre de ce travail, nous rappelons la théorie des représentants efficaces d'un élément pseudo-Anosov du mapping class group d'une surface S compacte orientée munie de n+1 points marqués. Ces objets ont été introduits par Bestvina-Handel et Los.<br /><br />Le deuxième chapitre contient l'exposé de la théorie des bons représentants et des représentants super efficaces d'un homéomorphisme pseudo-Anosov f fixant le point marqué x_0. Nous montrons ensuite un résultat de structure sur l'ensemble des représentants super efficaces : cet ensemble est une union d'un nombre fini de cycles qui sont parcourus en appliquant des opérations combinatoires. Nous en déduisons des algorithmes permettant de décider si l'homéomorphisme f - ou, ce qui est équivalent, sa classe d'isotopie - admet une racine fixant x_0, ou commute avec un élément d'ordre fini fixant x_0. Nous en déduisons également une nouvelle solution au problème de conjugaison parmi les éléments pseudo-Anosov du mapping class group qui fixent x_0.<br /><br />Dans le troisième chapitre, nous considérons un homéomorphisme f du disque et O une orbite de période n>=3 pour f. Nous donnons une minoration de l'entropie topologique des homéomorphismes isotopes à f relativement à O. Cette minoration est obtenue à l'aide de la théorie des représentants efficaces.<br /><br />Dans le quatrième chapitre, nous donnons des conditions nécessaires et suffisantes pour qu'une tresse beta à n brins admette une déstabilisation ou un mouvement d'échange. Ces conditions sont des propriétés sur l'élément du mapping class group induit par la tresse beta.