5 research outputs found

    Arithmetic Progressions and Chaos in Linear Dynamics

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    We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence of long arithmetic progressions in the sets of return times. We also show that this characterization does not hold for arbitrary Banach spaces. To achieve this, we study F-hypercyclicity for a family of subsets of the natural numbers associated to the existence of arbitrarily long arithmetic progressions.Fil: Cardeccia, Rodrigo Alejandro. Comisi贸n Nacional de Energ铆a At贸mica. Gerencia del 脕rea de Energ铆a Nuclear. Instituto Balseiro. Archivo Hist贸rico del Centro At贸mico Bariloche e Instituto Balseiro | Universidad Nacional de Cuyo. Instituto Balseiro. Archivo Hist贸rico del Centro At贸mico Bariloche e Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Centro Cient铆fico Tecnol贸gico Conicet - Patagonia Norte; ArgentinaFil: Muro, Luis Santiago Miguel. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Centro Cient铆fico Tecnol贸gico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas; Argentin

    Hypercyclic homogeneous polynomials on H(C)

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    It is known that homogeneous polynomials on Banach spaces cannot be hypercyclic, but there are examples of hypercyclic homogeneous polynomials on some non-normable Fr茅chet spaces. We show the existence of hypercyclic polynomials on H(C), by exhibiting a concrete polynomial which is also the first example of a frequently hypercyclic homogeneous polynomial on any F-space. We prove that the homogeneous polynomial on H(C) defined as the product of a translation operator and the evaluation at 0 is mixing, frequently hypercyclic and chaotic. We prove, in contrast, that some natural related polynomials fail to be hypercyclic.Fil: Cardeccia, Rodrigo. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Muro, Luis Santiago Miguel. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Centro Cient铆fico Tecnol贸gico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas; Argentin

    Orbits of homogeneous polynomials on Banach spaces

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    We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time-dense (the orbit meets every ball of radius), weakly dense and such that is dense for every that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.Fil: Cardeccia, Rodrigo Alejandro. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Oficina de Coordinaci贸n Administrativa Ciudad Universitaria. Instituto de Investigaciones Matem谩ticas "Luis A. Santal贸". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matem谩ticas "Luis A. Santal贸"; ArgentinaFil: Muro, Luis Santiago Miguel. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Centro Cient铆fico Tecnol贸gico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Informaci贸n y de Sistemas; Argentin

    Dynamics of homogeneous mappings

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    El objetivo de esta tesis es contribuir a la teor铆a de los sistemas din谩micos inducidos por una funci贸n homog茅nea en espacios de dimensi贸n infinita. Una aplicaci贸n S : X 鈫 X se dice hiperc铆clica si existe una 贸rbita densa en el espacio. Es sabido que ning煤n espacio de Banach admite un polinomio homog茅neo hiperc铆clico. Por otro lado se sabe que algunos espacios de Fr茅chet admiten polinomios homog茅neos hiperc铆clicos. Introducimos una noci贸n de conjunto de Julia asociado a un polinomio homog茅neo en un espacio de Banach. De las propiedades elementales de dicho conjunto concluimos que toda 贸rbita es nunca densa. Mostramos diversos ejemplos de polinomios homog茅neos que son hiperc铆clicos restringidos a su conjunto de Julia. En particular probamos que todo espacio de Banach, separable y de dimensi贸n infinita admite un polinomio homog脡neo distribucionalmente ca脫tico. Probamos que el polinomio e'1B . B es d茅bil hiperc铆clico, 未-hiperc铆clico y 螕-superc铆clico para todo 螕 膸 C no acotado y tal que 0 es punto de acumulaci贸n de 螕. M谩s a煤n estas propiedades son realizadas por la misma 贸rbita. Cabe destacar que ning煤n espacio de Banach admite un polinomio homog茅neo hiperc铆clico. Generalizamos la construcci贸n a espacios de Banach arbitrarios. Exhibimos el primer ejemplo de un polinomio homog茅neo hiperc铆clico en H(C), que es una fuente hist贸rica de ejemplos de operadores hiperc铆clicos. Probamos que el polinomio f 鈫 f(0).f(.+1) es mixing, ca贸tico y frecuentemente hiperc铆clico. Por otro lado mostramos que el polinomio f 鈫 f(0)f' no es hiperc铆clico. Respondemos una pregunta de B茅s y Conejero encontrando operadores bilineales hiperc铆clicos sin vectores hiperc铆clicos densos en X x X ... X. M谩s a煤n, exhibimos el primer operador bilineal hiperc铆clico en un espacio de Banach y probamos que todo espacio de Banach de dimensi贸n infinita y separable admite un operador bilineal hiperc铆clico. Respondemos una pregunta de Grosse Erdmann-Kim probando que todo espacio de Banach, separable y de dimensi贸n infinita admite un operador bilineal sim茅trico bihiperc铆clico. Estudiamos F-hiperc铆clicidad para dos familias de n煤meros naturales relacionadas con la existencia de progresiones aritm茅ticas arbitrariamente grandes. Introducimos las nociones de AP-hiperciclicidad y AP*-hiperciclicidad para operadores lineales. Mostramos que la noci贸n de AP-hiperc铆clididad es equivalente a que el operador sea hiperc铆clico y multiple recurrente. Proponemos un criterio simple de AP-hiperciclicidad que es implicado por el criterio fuerte de Kitai. Exhibimos un ejemplo de un operador que es AP-hiperc铆clico y no weakly mixing. Respondemos una pregunta de Costakis-Parisis probando que todo espacio de Banach admite un operador AP-hiperc铆clico. Probamos que para operadores 蠅*-蠅* continuos las nociones de AP*-hiperciclicidad, conjuntos peri贸dicos densos y caoticidad son equivalentes, respondiendo parcialmente una pregunta de Bonilla-Grosse Erdmann. Probamos que para backwardshifts las nociones de ser hiperc铆clico con conjuntos peri贸dicos densos y ca贸s son equivalentes. Mostramos un ejemplo de un backwards-hift en c0 que es AP*-hiperc铆clico pero no ca贸tico. Finalmente probamos que el espectro de los operadores AP*-hiperc铆clicos es perfecto.The aim of this thesis is to contribute to the theory of dynamical systems induced by a homogeneous mapping acting on an infinite dimensional space. A map S : X 鈫 X is called hypercyclic provided that there is a dense orbit in the space. It is well known that there are no hypercyclic homogeneous polynomials on Banach spaces. On the other hand some non normable Fr茅chet spaces are known to support hypercyclic homogeneous polynomials. We introduce the notion of Julia set associated to a homogeneous polynomial acting on a Banach space. From its basic properties we deduce that every orbit is nowhere dense. We give several examples of homogeneous polynomials being hypercyclic restricted to its Julia set. In particular we prove that every separable and infinite dimensional Banach space supports a distributionally chaotic homogeneous polynomial. We prove that the homogeneous polynomial e'1B on lp is weakly hypercyclic, 未-hypercyclic and 螕-supercyclic for every 螕 膸 C such that 螕 is either unbounded or such that 0 is an accumulation point of 螕. Moreover these properties are achieved by the same orbit. It is worth noticing that no homogeneous polynomial on a Banach space is hypercyclic. We generalize the construction to arbtrary infinite dimensional and separable Banach spaces. We exhibit the first example of a hypercyclic homogeneous polynomial on H(C), which is a historic source of examples of hypercyclic operators. We prove that the polynomial f 鈫 f(0).f(.+1) is mixing chaotic and frequently hypercyclic. On the other hand we show that the polynomial f 鈫 f(0)f' is not even hypercyclic. We answer a question due to B茅s and Conejero by finding examples of hypercyclic bilineal operators without a dense set of hypercyclic vectors. Moreover, we exhibit the first example of a hypercyclic bilineal operator on a Banach space. We generalize the construcion to arbitrary and separable Banach spaces. We answer a question of Grosse-Erdmann and Kim by proving that every separable and infinite dimensional Banach space supports a bihypercyclic symmetric bilineal operator. We study F-hypercyclicity for two families of natural numbers related to the existence of arbitrary long arithmetic progressions. We introduce the notion of AP-hypercyclicity and AP*-hypercyclicity for lineal operators. We prove that the concept of AP-hypercyclicity is equivalent to the operator being hypercyclic and multiple recurrent. We propose a simple criterion of AP-hypercyclicity which is implied by the strong Kitai criterion. We exhibit an example of an AP-hypercyclic operator which is weakly mixing. We answer a question of Costakis-Parisi by proving that every infinite dimensional and separable Banach space supports an AP-hypercyclic operator. We answer partially a question of Bonilla-Grosse Erdmann by proving that for 蠅*-蠅* continuous operators the notions of AP*-hypercyclicity, hypercyclicity with dense small periodic sets and chaocity are equivalent. We prove that for backwardshifts operators the concepts of having dense small periodic sets and being chaotic are equivalent. On the other hand we show an example of a non-chaotic backwardshift on c0 that is AP*-hypercyclic. We study the spectrum of AP*-hypercyclic operators and prove that they are perfect.Fil: Cardeccia, Rodrigo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

    Hypercyclic bilinear operators on Banach spaces

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    We study the dynamics induced by an m-linear operator. We answer a question of B猫s and Conejero showing an example of an m-linear hypercyclic operator acting on a Banach space. Moreover, we prove the existence of m-linear hypercyclic operators on arbitrary infinite dimensional separable Banach spaces. We also prove an existence result about symmetric bihypercyclic bilinear operators, answering a question by Grosse-Erdmann and Kim.Fil: Cardeccia, Rodrigo Alejandro. Consejo Nacional de Investigaciones Cient铆ficas y T茅cnicas. Oficina de Coordinaci贸n Administrativa Ciudad Universitaria. Instituto de Investigaciones Matem谩ticas "Luis A. Santal贸". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matem谩ticas "Luis A. Santal贸"; Argentin
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