UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS DEPARTAMENTO DE GEOMETRÍA Y TOPOLOGÍA ON THE DYNAMICS OF HOMEOMORPHISMS AROUND FIXED POINTS IN LOW DIMENSIÓN SOBRE LA DINÁMICA DE HOMEOMORFISMOS EN TORNO A PUNTOS FIJOS EN DIMENSIÓN BAJA TESIS DOCTORAL DE: LUIS HERNÁNDEZ CORBATO BAJO LA DIRECCIÓN DE: FRANCISCO ROMERO RUIZ DEL PORTAL Madrid, 2013 ©Luis Hernández Corbato, 2013 hola On the dynamics of homeomorphisms around fixed points in low dimension Sobre la din´amicade homeomorfismos en torno a puntos fijos en dimensi´onbaja Memoria presentada para optar al grado de Doctor en Ciencias Matem´aticaspor Luis Hern´andezCorbato Dirigida por D. Francisco Romero Ruiz del Portal Departamento de Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Universidad Complutense de Madrid Madrid, 2013 hola hola hola hola Agradecimientos Las primeras l´ıneasest´andedicadas a mi director de tesis, Paco. Le agradezco sus orientaciones y consejos, tanto los matem´aticos como los que se alejan de la ciencia, las innumerables discusiones en la pizarra y todo el empe~noque ha puesto en la tarea de guiarme durante estos a~nos.Su dedicaci´ony pasi´onpor las matem´aticasson ejemplares y han sido fundamentales para que este barco llegara a buen puerto. Me siento privilegiado por toda la atenci´ony el apoyo que me ha prestado. Es menester dar las gracias al Departamento de Geometr´ıay Topolog´ıay a la Fa- cultad de Matem´aticasde la Universidad Complutense. No s´olohan cubierto todas las necesidades b´asicasque un estudiante de doctorado puede precisar sino que me han per- mitido disfrutar de mis primeras experiencias como docente universitario. Al Ministerio le debo agradecer la beca FPU que me concedieron para realizar esta tesis y gracias a la cual he subsistido durante cuatro a~nos. Quiero agradecer a Patrice concederme la oportunidad de ir a Par´ıs,al Institut de Math´ematiquesde Jussieu. Sus grandes cualidades tanto humanas como matem´aticas han hecho que aprender y trabajar con ´elhaya sido una experiencia fant´astica. Mis estancias en Par´ıs bajo su tutela han sido claves para el desarrollo de la tesis. Mi tercera y ´ultimaestancia predoctoral, en R´ıo,fue tambi´enmuy enriquecedora. Desde aqu´ıdoy las gracias a Andr´esy a Enrique por su cercan´ıa,disposici´ony voluntad. Creo que es buen momento para recordar a quienes estuvieron involucrados en mis primeros e inocentes pasos dentro de esta ciencia, hace ya algunos a~nos.Por ello, adem´as de a mis padres, quiero expresar mi gratitud, entre otros, a Mar´ıa,Josep, Marco, Merche y Joaqu´ın. Gracias a todos mis amigos por los muchos buenos ratos que hemos disfrutado durante este tiempo, en las situaciones m´asvariopintas: molestando en alg´undespacho, en los seminarios de lectura, compartiendo un caf´ede superh´eroe, en un agujero de gusano, viendo un partido, tomando unas raciones, unas ca~nas,unas pizzas, jugando a la pocha o a lo que tocara o simplemente comentando la jugada. Menci´onespecial para aquellos que est´ano han estado fuera, y que en sus visitas me han acompa~nado.Es precisamente lejos de casa cuando m´asse agradece la compa~n´ıay por eso quiero aqu´ıacordarme de todos los buenos amigos con los que compart´ıvivencias tanto en Par´ıscomo en R´ıo,gracias a los cuales se me hicieron m´asllevaderas las estancias. A mis padres y hermano, gracias por vuestra paciencia, comprensi´ony por estar siempre ah´ıcuando he necesitado ayuda. Gracias tambi´ena mi t´ıay abuelas por vuestro cari~no. Tengo la suerte de haber podido recorrer este largo camino muy bien acompa~nado. Laura, gracias por soportarme y por tu apoyo incondicional. Contents Summary i Resumen xi 1 Rotation numbers for planar attractors1 1.1 Introduction...................................1 1.2 Preliminaries..................................4 1.2.1 Rotation number............................4 1.2.2 Denjoy homeomorphisms.......................5 1.2.3 Prime end theory............................8 1.3 Setting and previous results.......................... 10 1.4 Example of maps with irrational rotation number.............. 11 1.5 Irrational rotations do not appear on prime ends.............. 19 1.6 About the existence of periodic points.................... 23 1.6.1 Partial converse result......................... 23 1.6.2 Counterexample to the conjecture.................. 26 2 Fixed point index and the Conley index 31 2.1 Introduction................................... 31 2.1.1 Algebraic invariants in dynamical systems.............. 31 2.1.2 Statement of the results........................ 33 2.1.3 Fixed point index results....................... 39 2.2 Preliminaries.................................. 40 2.2.1 Fixed point index............................ 40 2.2.2 Shift equivalence............................ 43 2.2.3 Permutation endomorphisms..................... 46 2.2.4 Acyclic continua............................ 48 2.2.5 Isolating blocks and filtration pairs.................. 49 2.2.6 Conley index and Lefschetz-Dold Theorem............. 52 2.2.7 Attractors, repellers and nilpotence.................. 54 2.3 Fixed point index for continuous maps in R2 ................ 57 2.3.1 Statement of the theorem....................... 57 2.3.2 Splitting and nilpotence arguments.................. 59 2.3.3 Characterization of fixed point index sequences........... 61 2.3.4 Realization of all possible sequences................. 64 2.4 Some questions around discrete Conley index theory............ 65 2.4.1 Around the definition......................... 65 2.4.2 Another approach to the discrete Conley index........... 71 2.4.3 On the connectedness of N L .................... 73 n 2.4.4 Duality................................. 77 2.5 Main results................................... 81 2.5.1 Statement of the key theorem and proofs.............. 81 2.5.2 A toy model: a radial case....................... 83 n 2.5.3 Realizing all possible sequences i(f ; p) n≥1 ............ 85 f g 2.5.4 Proof I: Construction of a suitable pair............... 87 2.5.5 Proof II: Nilpotence.......................... 89 2.5.6 Proof III: Definition of ', description of f∗;1 ............. 90 2.5.7 Proof IV: Introducing the stable set................. 91 2.5.8 Proof V: Gathering all dynamical informatione closer to the unsta- ble set.................................. 92 2.5.9 Proof VI: Higher-dimensional homological decomposition..... 93 2.6 Further remarks................................ 95 2.7 Local zeta function............................... 97 Summary Introduction The work on celestial mechanics by Henri Poincar´ein the end of the 19th century was the starting point of the theory of dynamical systems. Even though in its origin it was developed as a branch of the study of differential equations, this theory quickly gained interest by itself. On the one hand, there exist a great amount of evolutive processes whose description does not adjust to the language of differential equations, though the notion of dynamical system is broad enough to encompass them. On the other hand, analytical methods developed to study differential equations proved to have limited scope and geometrical and topological techniques appeared out of necessity for the study of dynamics. During the 20th century, the work of renowned mathematicians contributed to raise the importance of dynamical systems and made connections with other areas of mathematics. This work belongs to the area of topological dynamics, as most of our discussions are purely topological. As a rule of thumb, dynamics subject of study are not smooth, nor we care about whether they preserve a measure. The type of evolution followed by our systems is discrete and time independent. In other words, we consider autonomous dynamical systems generated by the iteration of a map. As one of the simplest possible kinds of evolution, it provides more freedom for the dynamics and, in return, more difficulties for its study. Although the title of our work is vague on purpose, we restrict virtually all our discussion to dimensions 2 and 3, where topological obstructions prevent the existence of certain discrete dynamics. In dimension 1, discrete dynamical systems have trivial descriptions as long as the generating map is invertible or, equivalently, they are time reversible. Otherwise, their behavior may be very complicated, as an example see the bifurcation diagram of the quadratic family of maps in the unit interval. There is a wide literature dealing with dynamics generated surfaces homeomorphisms. Several techniques which are specific to dimension 2 have been successfully applied to obtain powerful results. Perhaps one of the first results in this direction is the so-called Brouwer's Lemma of translation arcs [Br12, Br84]. It states that for planar orientation-preserving homeomorphisms, the existence of a recurrent point implies that of a fixed point. In particular, any periodic orbit \encloses" a fixed point, much alike an invariant circle bounds a closed disk which always contains a fixed point, by Brouwer's fixed point Theorem. Prime end theory, used in the first chapter of this dissertation, is another example of a planar technique employed in the study of surface dynamics. Understanding the dynamics, even in the case of flows, becomes much more difficult in dimension 3 and higher, as the topology of i ii the phase space does not pose so many restrictions. The task of studying a particular dynamical system in great detail is typically twofold. First, a careful selection of pieces of the dynamics are examined locally. Then, one has to specify how these pieces are glued to compose the global picture. Some of the great contributions by Morse, Smale or Conley to dynamical systems share this motivation. For the local study, typically one starts with neighborhoods of fixed points, periodic orbits or other minimal sets, as they are dynamically indecomposable pieces. For example, in the discrete case, hyperbolic fixed points of diffeomorphisms have trivial local dynamics, as Hartman-Grobman's Theorem implies the map is locally conjugate to a linear map in a neighborhood of the fixed point. However, in a more general setting the task of classifying local dynamics often becomes hard, either because of a rich structure of the sets (e.g.