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Dissertations / Theses on the topic 'Ergodic theory'
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Quas, Anthony Nicholas. "Some problems in ergodic theory." Thesis, University of Warwick, 1993. http://wrap.warwick.ac.uk/58569/.
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The thesis consists of a study of problems in ergodic theory relating to one-dimensional dynamical systems, Markov chains and generalizations of Markov chains. It is divided into chapters, three of which have appeared in the literature as papers. Chapter 1 looks at continuous families of circle maps and investigates conditions under which there is a weak*-continuous family of invariant measures. Sufficient conditions are exhibited and the necessity of these conditions is investigated. Chapter 2 is about expanding maps of the interval and the circle, and their relation with g-measures and generalized baker's transformations. The g-measures are generalizations of Markov chains to stochastic processes with infinite memory and generalized baker's transformations are geometric realizations of these. The chapter is based around the question of whether there exist expanding maps preserving Lebesgue measure, for which Lebesgue measure is not ergodic. Results are known if the map is sufficiently differentiable (for example C1+α), but the C1 case is still unclear. The chapter contains some partial solutions to this question. Chapter 3 is about representation of Markov chains on compact manifolds by measured collections of smooth maps. Given a measured collection of maps, a Markov chain is induced in a natural fashion. This chapter is about reversing this process. Chapter 4 describes a specialization of the setting of Chapter 3 to Markov chains on tori. In this case, it is possible to demand more of the maps of the representation than smoothness. In particular, they can be chosen to be local diffeomorphisms. The chapter also addresses the question of whether in general the maps can be taken to be diffeomorphisms and gives a counterexample showing that there exist Markov chains on tori which do not admit a representation by diffeomorphisms.
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Bulinski, Kamil. "Interactions between Ergodic Theory and Combinatorial Number Theory." Thesis, The University of Sydney, 2017. http://hdl.handle.net/2123/17733.
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The seminal work of Furstenberg on his ergodic proof of Szemerédi’s Theorem gave rise to a very rich connection between Ergodic Theory and Combinatorial Number Theory (Additive Combinatorics). The former is concerned with dynamics on probability spaces, while the latter is concerned with Ramsey theoretic questions about the integers, as well as other groups. This thesis further develops this symbiosis by establishing various combinatorial results via ergodic techniques, and vice versa. Let us now briefly list some examples of such. A shorter ergodic proof of the following theorem of Magyar is given: If B Zd, where d 5, has upper Banach density at least > 0, then the set of all squared distances in B, i.e., the set fkb1 b2k2 j b1; b2 2 Bg, contains qZ>R for some integer q = q( ) > 0 and R = R(B). Our technique also gives rise to results on the abundance of many other higher order Euclidean configurations in such sets. Next, we turn to establishing analogues of this result of Magyar, where k k2 is replaced with other quadratic forms and various other algebraic functions. Such results were initially obtained by Björklund and Fish, but their techniques involved some deep measure rigidity results of Benoist-Quint. We are able to recover many of their results and prove some completely new ones (not obtainable by their techniques) in a much more self-contained way by avoiding these deep results of Benoist-Quint and using only classical tools from Ergodic Theory. Finally, we extend some recent ergodic analogues of the classical Plünnecke inequalities for sumsets obtained by Björklund-Fish and establish some estimates of the Banach density of product sets in amenable non-abelain groups. We have aimed to make this thesis accesible to readers outside of Ergodic Theory who may be primarily interested in the arithmetic and combinatorial applications.
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Butkevich, Sergey G. "Convergence of Averages in Ergodic Theory." The Ohio State University, 2001. http://rave.ohiolink.edu/etdc/view?acc_num=osu980555965.
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Butkevich, Sergey. "Convergence of averages in Ergodic Theory /." The Ohio State University, 2000. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488196781735316.
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Johnson, Bryan R. "Unconditional convergence of differences in ergodic theory /." The Ohio State University, 1997. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487945015615412.
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Jaššová, Alena. "On ergodic theory in non-Archimedean settings." Thesis, University of Liverpool, 2014. http://livrepository.liverpool.ac.uk/2006322/.
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Meco, Benjamin. "Ergodic Theory and Applications to Combinatorial Problems." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-409810.
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Prabaharan, Kanagarajah. "Topics in ergodic theory : existence of invariant elements and ergodic decompositions of Banach lattices /." The Ohio State University, 1991. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487688973685025.
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Cannizzo, Jan. "Schreier Graphs and Ergodic Properties of Boundary Actions." Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31444.
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This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.
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Raissi-Dehkordi, Ramin. "Ergodic theory of dynamical systems having absolutely continuous spectrum." Thesis, University of Cambridge, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.627274.
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Li, Kuo-tung. "Convergence problems arising from harmonic analysis and ergodic theory /." The Ohio State University, 1994. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487853913100673.
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Snyman, Mathys Machiel. "Ergodic properties of noncommutative dynamical systems." Diss., University of Pretoria, 2013. http://hdl.handle.net/2263/40351.
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In this dissertation we develop aspects of ergodic theory
for C*-dynamical systems for which the C*-algebras are allowed
to be noncommutative. We define four ergodic properties,
with analogues in classic ergodic theory, and study C*-dynamical
systems possessing these properties. Our analysis will show that, as
in the classical case, only certain combinations of these properties
are permissable on C*-dynamical systems. In the second half of
this work, we construct concrete noncommutative C*-dynamical
systems having various permissable combinations of the ergodic
properties. This shows that, as in classical ergodic theory, these
ergodic properties continue to be meaningful in the noncommutative
case, and can be useful to classify and analyse C*-dynamical
systems.Dissertation (MSc)--University of Pretoria, 2013.
gm2014
Mathematics and Applied Mathematics
unrestricted
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Beyers, Frederik J. C. "A Hilbert space approach to multiple recurrence in ergodic theory." Pretoria : [s.n.], 2004. http://upetd.up.ac.za/thesis/available/etd-02222006-104936.
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Wierdl, Mate. "Almost everywhere convergence and recurrence along subsequences in ergodic theory /." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487672631600881.
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Rothlisberger, Matthew Samuel. "Ergodic and Combinatorial Proofs of van der Waerden's Theorem." Scholarship @ Claremont, 2010. http://scholarship.claremont.edu/cmc_theses/14.
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Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.
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King, Malcolm Bruce. "Joinings and relative ergodic properties of W*-dynamical systems." Thesis, University of Pretoria, 2019. http://hdl.handle.net/2263/73237.
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We prove a characterization of relative weak mixing in W*-dynamical systems in terms of a relatively independent joining. We then define a noncommutative version of relative discrete spectrum, show that it generalizes
both the classical and noncommutative absolute cases and give examples.
Chapter 1 reviews the GNS construction for normal states, the related
semicyclic representation on von Neumann algebras, Tomita-Takasaki theory and conditional expectations. This will allow us to define, in the tracial case, the basic construction of Vaughan Jones and its associated lifted
trace. Dynamics is introduced in the form of automorphisms on von Neumann algebras, represented using the cyclic and separating vector and then
extended to the basic construction.
In Chapter 2, after introducing a relative product system, we discuss
relative weak mixing in the tracial case. We give an example of a relative
weak mixing W*-dynamical system that is neither ergodic nor asymptotically abelian, before proving the aforementioned characterization.
Chapter 3 defines relative discrete spectrum as complementary to relative weak mixing. We motivate the definition using work from Chapter 2.
We show that our definition generalizes the classical and absolute noncommutative case of isometric extensions and discrete spectrum, respectively.
The first example is a skew product of a classical system with a noncommutative one. The second is a purely noncommutative example of a tensor
product of a W*-dynamical system with a finite-dimensional one.Thesis (PhD)--University of Pretoria, 2019.
Pilot Programme Top-Up Bursary, Department of Mathematics and Applied Mathematics, University of Pretoria.
Mathematics and Applied Mathematics
PhD
Unrestricted
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Rehacek, Jan. "Ergodic billiards and mechanism of defocusing in N dimensions." Diss., Georgia Institute of Technology, 1996. http://hdl.handle.net/1853/28886.
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Beyers, Frederik Johannes Conradie. "The Szemeredi property in noncommutative dynamical systems." Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-05242009-145506.
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Kunde, Philipp [Verfasser], and Reiner [Akademischer Betreuer] Lauterbach. "Combinatorial constructions in Smooth Ergodic Theory / Philipp Kunde. Betreuer: Reiner Lauterbach." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2014. http://d-nb.info/1064077056/34.
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Munday, Sara Ann. "Finite and infinite ergodic theory for linear and conformal dynamical systems." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/3220.
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The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group.
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Donoso, Sebastian Andres. "Contributions to ergodic theory and topological dynamics : cube structures and automorphisms." Thesis, Paris Est, 2015. http://www.theses.fr/2015PEST1007/document.
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Cette thèse est consacrée à l'étude des différents problèmes liés aux structures des cubes , en théorie ergodique et en dynamique topologique. Elle est composée de six chapitres. La présentation générale nous permet de présenter certains résultats généraux en théorie ergodique et dynamique topologique. Ces résultats, qui sont associés d'une certaine façon aux structures des cubes, sont la motivation principale de cette thèse. Nous commençons par les structures de cube introduites en théorie ergodique par Host et Kra (2005) pour prouver la convergence dans $L^2 $ de moyennes ergodiques multiples. Ensuite, nous présentons la notion correspondante en dynamique topologique. Cette théorie, développée par Host, Kra et Maass (2010), offre des outils pour comprendre la structure topologique des systèmes dynamiques topologiques. En dernier lieu, nous présentons les principales implications et extensions dérivées de l'étude de ces structures. Ceci nous permet de motiver les nouveaux objets introduits dans la présente thèse, afin d'expliquer l'objet de notre contribution. Dans le Chapitre 1, nous nous attachons au contexte général en théorie ergodique et dynamique topologique, en mettant l'accent sur l'étude de certains facteurs spéciaux. Les Chapitres 2, 3, 4 et 5 nous permettent de développer les contributions de cette thèse. Chaque chapitre est consacré à un thème particulier et aux questions qui s'y rapportent, en théorie ergodique ou en dynamique topologique, et est associé à un article scientifique. Les structures de cube mentionnées plus haut sont toutes définies pour un espace muni d'une unique transformation. Dans le Chapitre 2, nous introduisons une nouvelle structure de cube liée à l'action de deux transformations S et T qui commutent sur un espace métrique compact X. Nous étudions les propriétés topologiques et dynamiques de cette structure et nous l'utilisons pour caractériser les systèmes qui sont des produits ou des facteurs de produits. Nous présentons également plusieurs applications, comme la construction des facteurs spéciaux. Le Chapitre 3 utilise la nouvelle structure de cube définie dans le Chapitre 2 dans une question de théorie ergodique mesurée. Nous montrons la convergence ponctuelle d'une moyenne cubique dans un système muni deux transformations qui commutent. Dans le Chapitre 4, nous étudions le semigroupe enveloppant d'une classe très importante des systèmes dynamiques, les nilsystèmes. Nous utilisons les structures des cubes pour montrer des liens entre propriétés algébriques du semigroupe enveloppant et les propriétés topologiques et dynamiques du système. En particulier, nous caractérisons les nilsystèmes d'ordre 2 par une propriété portant sur leur semigroupe enveloppant. Dans le Chapitre 5, nous étudions les groupes d'automorphismes des espaces symboliques unidimensionnels et bidimensionnels. Nous considérons en premier lieu des systèmes symboliques de faible complexité et utilisons des facteurs spéciaux, dont certains liés aux structures de cube, pour étudier le groupe de leurs automorphismes. Notre résultat principal indique que, pour un système minimal de complexité sous-linéaire, le groupe d'automorphismes est engendré par l'action du shift et un ensemble fini. Par ailleurs, en utilisant les facteurs associés aux structures de cube introduites dans le Chapitre 2, nous étudions le groupe d'automorphismes d'un système de pavages représentatif. La bibliographie, commune à l'ensemble de la thèse, se trouve en fin documentThis thesis is devoted to the study of different problems in ergodic theory and topological dynamics related to og cube structures fg. It consists of six chapters. In the General Presentation we review some general results in ergodic theory and topological dynamics associated in some way to cubes structures which motivates this thesis. We start by the cube structures introduced in ergodic theory by Host and Kra (2005) to prove the convergence in $L^2$ of multiple ergodic averages. Then we present its extension to topological dynamics developed by Host, Kra and Maass (2010), which gives tools to understand the topological structure of topological dynamical systems. Finally we present the main implications and extensions derived of studying these structures, we motivate the new objects introduced in the thesis and sketch out our contributions. In Chapter 1 we give a general background in ergodic theory and topological dynamics given emphasis to the treatment of special factors. % We give basic definitions and describe special factors associated to a From Chapter 2 to Chapter 5 we develop the contributions of this thesis. Each one is devoted to a different topic and related questions, both in ergodic theory and topological dynamics. Each one is associated to a scientific article. In Chapter 2 we introduce a novel cube structure to study the actions of two commuting transformations $S$ and $T$ on a compact metric space $X$. In the same chapter we study the topological and dynamical properties of such structure and we use it to characterize products systems and their factors. We also provide some applications, like the construction of special factors. In the same topic, in Chapter 3 we use the new cube structure to prove the pointwise convergence of a cubic average in a system with two commuting transformations. In Chapter 4, we study the enveloping semigroup of a very important class of dynamical systems, the nilsystems. We use cube structures to show connexions between algebraic properties of the enveloping semigroup and the geometry and dynamics of the system. In particular, we characterize nilsystems of order 2 by its enveloping semigroup. In Chapter 5 we study automorphism groups of one-dimensional and two-dimensional symbolic spaces. First, we consider low complexity symbolic systems and use special factors, some related to the introduced cube structures, to study the group of automorphisms. Our main result states that for minimal systems with sublinear complexity such groups are spanned by the shift action and a finite set. Also, using factors associated to the cube structures introduced in Chapter 2 we study the automorphism group of a representative tiling system. The bibliography is defer to the end of this document
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Donoso, Fuentes Sebastián Andrés. "Contributions to ergodic theory and topological dynamics: cube structures and automorphisms." Tesis, Universidad de Chile, 2015. http://repositorio.uchile.cl/handle/2250/135055.
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Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEsta tesis está consagrada al estudio de diferentes problemas en teoría ergódica y dinámica topológica, relacionados a "estructuras de cubos". Consta de seis capítulos. En la presentación general entregamos resultados generales, ligados en cierta manera a las estructuras de cubos que motivan esta tesis. Comenzamos por las estructuras de cubos introducidas en teoría ergódica por Host y Kra para probar la convergencia en L^2 de medias ergódicas múltiples. Luego presentamos su extensión a dinámica topológica, desarrollada por Host, Kra y Maass (2010), que entrega herramientas para entender la estructura topológica de sistemas dinámicos topológicos. Finalmente, mostramos las implicancias y extensiones principales derivadas de estudiar estas estructuras, motivamos los nuevos objetos introducidos en esta tesis y bosquejamos nuestras contribuciones. En el Capítulo 1, entregamos antecedes generales en teoría ergódica y dinámica topológica, dando énfasis al estudio de ciertos factores especiales. Desde el Capítulo 2 al Capítulo 5 desarrollamos las contribuciones de esta tesis. Cada uno está consagrado a un tópico diferente y a sus problemáticas relacionadas, tanto en teoría ergódica como en dinámica topológica. Cada uno está asociado a un artículo científico. En el Capítulo 2 introducimos una nueva estructura de cubos para estudiar la acción de dos transformaciones S y T que conmutan, sobre un espacio métrico compacto X. En el mismo capítulo estudiamos las propiedades topológicas y dinámicas de tales estructuras y las usamos para caracterizar productos de sistemas y sus factores. También damos algunas aplicaciones, como la construcción de factores especiales. En el mismo tema, en el Capítulo 3 usamos esta nueva estructura para probar la convergencia casi segura de una media cúbica en un sistema con dos transformaciones que conmutan. En el Capítulo 4, estudiamos el semigrupo envolvente de una clase importante de sistemas dinámicos, los nilsistemas. Usamos estructuras de cubos para mostrar relaciones entre propiedades algebraicas del semigrupo envolvente con la geometría y dinámica de un sistema. En particular, caracterizamos nilsistemas de orden 2 vía el semigrupo envolvente. En el Capítulo 5 estudiamos grupos de automorfismos de sistemas simbólicos uno y dos dimensionales. Primero consideramos sistemas simbólicos de baja complejidad y usamos factores especiales, algunos ligados a estructuras de cubos, para estudiar el grupo de automorfismos. Nuestro resultado principal establece que en sistemas minimales de complejidad sublineal, tales grupos son generados por el shift y un conjunto finito. También, usando factores asociados a las estructuras de cubos del Capítulo 2, estudiamos el grupo de automorfismos de un sistema de embaldosados representativo. Las referencias bibliográficas aparecen al final del documento.
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Kunde, Philipp Verfasser], and Reiner [Akademischer Betreuer] [Lauterbach. "Combinatorial constructions in Smooth Ergodic Theory / Philipp Kunde. Betreuer: Reiner Lauterbach." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2014. http://d-nb.info/1064077056/34.
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Artigiani, Mauro. "Some aspects of diophantine approximation and ergodic theory of translation surfaces." Thesis, University of Bristol, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.702866.
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This thesis deals with two different topics. In the first part we study Lagrange spectra of Veech translation surfaces, which are a generalisation of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. We start from the concrete example given by the surface obtained glueing a regular octagon. We use the coding developed by Smillie and Ulcigrai for the surfaces obtained glueing the regular 2ɳ-gons to code geodesics in the Teichmiiller disk of the octagon and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets. In particular this yields an estimate on the beginning point of the Hall ray. Generalising the approach of the octagon, in a joint work with Luca Marchese and Corinna Ulcigrai, we prove the existence of a Hall ray for the Lagrange spectrum of any Veech translation surface. In this case, we use the boundary expansion developed by Bowen and Series. In the second part, we construct exceptional examples of ergodic vertical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are Z²-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed. iii
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Mortiss, Genevieve. "Average co-ordinate entropy and a non-singular version of restricted orbit equivalence." [Sydney : University of New South Wales], 1997. http://www.library.unsw.edu.au/~thesis/adt-NUN/public/adt-NUN1998.0001/index.html.
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Shi, Ronggang. "Equidistribution of expanding measures with local maximal dimension and Diophantine Approximation." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1242259439.
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Kaimanovich, Vadim, Klaus Schmidt, and Klaus Schmidt@univie ac at. "Ergodicity of cocycles. 1: General Theory." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi936.ps.
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Tanzi, Matteo. "Heterogeneously coupled maps : from high to low dimensional systems through ergodic theory." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/50709.
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In this thesis we study ergodic theoretical properties of high-dimensional systems coupled on graphs. The local dynamics at each node is hyperbolic and coupled with other nodes according to the edges of the graph. We focus our attention on the case of graphs with heterogeneous degrees meaning that most of the nodes make a small number of interactions, while a few hub nodes have very high degree. For such high-dimensional systems there is a regime of the interaction strength for which the coupling is small for poorly connected systems, and large for the hub nodes. In particular, global hyperbolicity might be lost. We show that, under certain hypotheses, the dynamics of hub nodes can be very well approximated by a low-dimensional system for exponentially long time in the size of the network and that the system exhibit hyperbolic behaviour in this time window. Even if this describes only a long transient, we argue that this is the behaviour that one expects to observe in experiments. Such a description allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections). The HCM we study provide a new paradigm to explain why and how the dynamics of a network dynamical system can change across layers.
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Griesmer, John Thomas. "Ergodic averages, correlation sequences, and sumsets." Columbus, Ohio : Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view.cgi?acc%5Fnum=osu1243973834.
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Jarrett, Kieran. "Non-singular actions of countable groups." Thesis, University of Bath, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761021.
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In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.
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Naughton, David Vincent. "On the fine structure of dynamically-defined invariant graphs." Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/on-the-fine-structure-of-dynamicallydefined-invariant-graphs(0e81b14b-af68-4433-a983-f98a7b8c5d86).html.
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MENDICO, CRISTIAN. "Ergodic behavior of control systems and first-order mean field games." Doctoral thesis, Gran Sasso Science Institute, 2021. http://hdl.handle.net/20.500.12571/23542.
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The work in this thesis concerns the analysis of first-order mean field game (MFG) systems with control of acceleration and the study of the long time-average behavior of control systems of sub-Riemannian type. More precisely, in the first part we begin by studying the well-posedness of the MFG system associated with a control problem with linear state equation. In particular, via a relaxed approach, we prove the existence and the uniqueness of mild solutions and we also study their regularity. Then, we focus on the MFG system with control of the acceleration, a particular case of the one above, and we investigate the long time-average behavior of solutions showing the convergence to the critical constant. Here, as for the previous analysis, the main issues are the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable. Indeed, for instance, when studying the asymptotic behavior of the control system this lead us to a non existence result of continuous viscosity solutions to the ergodic Hamilton-Jacobi equation. Consequently, it does not allowed us to the define the ergodic MFG system as one would expect. We conclude this first part establishing a connection between the MFG system with control of acceleration and the classical one. To do so, we study the singular perturbation problem for MFG system of acceleration, that is, we analyze the behavior of solutions to the system when the acceleration cost goes to zero. Again, we solve the problem by using variation techniques due to the problems arising from the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable. In the second part, we concentrate the attention to drift-less affine control systems (sub-Riemannian type). Differently from the case of acceleration, we prove that there exists a critical constant and the ergodic Hamilton-Jacobi equation associated with such a constant has continuous viscosity solutions. This is possible appealing to the properties of the sub-Riemannian geometry on the state space. Still using the properties of this geometry we finally define the Lax-Oleinink semigroup and we prove the existence of a fixed point of such semigroup. We conclude this part, and thus this thesis, extending the celebrated Aubry-Mather Theory to the case of sub-Riemannian control system. We first show a variational representation formula for the critical constant and from this we define the Aubry set. By using a dynamical approach we study the analytical and topological properties of such sets as, for instance, horizontal differentiability of the critical solution at any points lying in such a set.
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Shereshevsky, Mark Alexandrovich. "On the ergodic theory of cellular automata and two-dimensional Markov shifts generated by them." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/34640/.
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In this thesis we study measurable and topological dynamics of certain classes of cellular automata and multi-dimensional subshifts. In Chapter 1 we consider one-dimensional cellular automata, i.e. the maps T: PZ -> PZ (P is a finite set with more than one element) which are given by (Tx)i==F(xi+1, ..., xi+r), x=(xi)iEZ E PZ for some integers 1≤r and a mapping F: Pr-1+1 -> P. We prove that if F is right- (left-) permutative (in Hedlund's terminology) and 0≤1
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Mortiss, Genevieve Catherine Mathematics UNSW. "Average co-ordinate entropy and a non-singular version of restricted orbit equivalence." Awarded by:University of New South Wales. Mathematics, 1997. http://handle.unsw.edu.au/1959.4/17823.
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A notion of entropy is defined for the non-singular action of finite co-ordinate changes on X - the infinite product of two- point spaces. This quantity - average co-ordinate or AC entropy - is calculated for product measures and G-measures on X, and an equivalence relation is established for which AC entropy is an invariant. The Inverse Vitali Lemma is discussed in a measure preserving context, and it is shown that for a certain class of measures on X known as odometer bounded, the result will still hold for odometer actions. The foundations for a non-singular version of Rudolph's restricted orbit equivalence are established, and a size for non-singular orbit equivalence is introduced. It is shown that provided the Inverse Vitali Lemma still holds, the non-singular orbit equivalence classes can be described using this new size.
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Pavlov, Ronald Lee. "Some results on recurrence and entropy." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1180454690.
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Kadyrov, Shirali. "Entropy and Escape of Mass in Non-Compact Homogeneous Spaces." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1272027404.
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Grigo, Alexander. "Billiards and statistical mechanics." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29610.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.Committee Chair: Bunimovich, Leonid; Committee Member: Bonetto, Federico; Committee Member: Chow, Shui-Nee; Committee Member: Cvitanovic, Predrag; Committee Member: Weiss, Howard. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Szkola, Arleta. "The Shannon-McMillan theorem and related results for ergodic quantum spin lattice systems and applications in quantum information theory." [S.l.] : [s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=970619588.
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Udrea, Bogdan Teodor. "Applications of deformation rigidity theory in Von Neumann algebras." Diss., University of Iowa, 2012. https://ir.uiowa.edu/etd/3395.
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This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.
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Andrade, Rodrigo Manoel Dias [UNESP]. "Bilhares planares." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/92952.
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O objetivo principal deste trabalho e estudar a dinâmica de uma partícula pontual no interior de subconjuntos do plano. Tais sistemas são conhecidos na literatura como bilhares. Apresentaremos os principais conceitos desses sistemas e veremos que tais sistemas deixam invariante uma medida de probabilidade, o que nos permite aplicar a Teoria Ergódica ao problema do bilhar
The main goal of this work is to study the dynamical behavior of a point-like (dimensionless) particle in the interior of planar regions. Such systems are known in the literature as billiards. We're going to present the principal concepts of those systems and we'll see that such system turns the probability measure invariant, which allows us to apply the Ergodic Theory to billiard problems
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Karagulyan, Davit. "Certain results on the Möbius disjointness conjecture." Doctoral thesis, KTH, Matematik (Inst.), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-215682.
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We study certain aspects of the Möbius randomness principle and more specifically the Möbius disjointness conjecture of P. Sarnak. In paper A we establish this conjecture for all orientation preserving circle homeomorphisms and continuous interval maps of zero entropy. In paper B we show, that for all subshifts of finite type with positive topological entropy the Möbius disjointness does not hold. In paper C we study a class of three-interval exchange maps arising from a paper of Bourgain and estimate its Hausdorff dimension. In paper D we consider the Chowla and Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationship.QC 20171016
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Alves, Fabricio Fernando [UNESP]. "Teorema ergódico multiplicativo de oseledets." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/94212.
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Este trablaho apresenta os conceitos de Lyapounov e de espaços próprios e fornece um resultado devido a Oseledets, o qual trata da existência desses expoentes (e, consequentemente, dos espaços próprios) do ponto de vista da teoria da medida. A prova do teorema que nós fornecemos foi dada originalmente por Mañe e posteriormente melhorada por Viana.
This work presents the concepts of Lyapounov exponents and of proper spaces and provides a result due to Oseledets, wich deals with the existence of these exponents (and consequently, of the proper spaces) from a measure-theoretical point of view. The proof of the theorem which we provide was originally given by Mañe later improved by Viana.
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Samuel, Anthony. "A commutative noncommutative fractal geometry." Thesis, University of St Andrews, 2010. http://hdl.handle.net/10023/1710.
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In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.
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Bhattad, Kapil. "Joint source channel coding for non-ergodic channels: the distortion signal-to-noise ratio (SNR) exponent perspective." Texas A&M University, 2008. http://hdl.handle.net/1969.1/85928.
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We study the problem of communicating a discrete time analog source over
a channel such that the resulting distortion is minimized. For ergodic channels,
Shannon showed that separate source and channel coding is optimal. In this work we
study this problem for non-ergodic channels.
Although not much can be said about the general problem of transmitting any
analog sources over any non-ergodic channels with any distortion metric, for many
practical problems like video broadcast and voice transmission, we can gain insights
by studying the transmission of a Gaussian source over a wireless channel with mean
square error as the distortion measure. Motivated by different applications, we consider three different non-ergodic channel models - (1) Additive white Gaussian noise
(AWGN) channel whose signal-to-noise ratio (SNR) is unknown at the transmitter; (2)
Rayleigh fading multiple-input multiple-output MIMO channel whose SNR is known
at the transmitter; and (3) Rayleigh fading MIMO channel whose SNR is unknown
at the transmitter.
The traditional approach to study these problems has been to fix certain SNRs
of interest and study the corresponding achievable distortion regions. However, the
problems formulated this way have not been solved even for simple setups like 2
SNRs for the AWGN channel. We are interested in performance over a wide range
of SNR and hence we use the distortion SNR exponent metric to study this problem.
Distortion SNR exponent is defined as the rate of decay of distortion with SNR in the high SNR limit.
We study several layered transmissions schemes where the source is first compressed in layers and then the layers are transmitted using channel codes that provide
variable error protection. Results show that in several cases such layered transmission
schemes are optimal in terms of the distortion SNR exponent. Specifically, if the band-
width expansion (number of channel uses per source sample) is b, we show that the
optimal distortion SNR exponent for the AWGN channel is b and it is achievable using
a superposition based layered scheme. For the L-block Rayleigh fading M x N MIMO
channel the optimal exponent is characterized for b < (|N - M|+1)= min(M;N) and
b > MNL2. This corresponds to the entire range of b when min(M;N) = 1 and
L = 1. The results also show that the exponents obtained using layered schemes
which are a small subclass of joint source channel coding (JSCC) schemes are, surprisingly, as good as and better in some cases than achievable exponent of all other
JSCC schemes reported so far.
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Figueiredo, Fernanda Ronssani de. "Medidas que maximizam a entropia no Deslocamento de Haydn." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2015. http://hdl.handle.net/10183/127974.
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Neste trabalho é abordado o exemplo proposto por Nicolai Haydn, no qual é dado um exemplo de um deslocamento onde é possível construir in nitas medidas de máxima entropia, além de in nitos estados de equilíbrio.In this work, we present the example shown by Nicolai Haydn, which is given by subshift where is possible to show in nity measures of maximal entropy, besides in nitely many distinct equilibrium states.
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Nasca, Angelo J. III. "The Linear Dynamics of Several Commuting Operators." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1420228736.
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Alves, Fabricio Fernando. "Teorema ergódico multiplicativo de oseledets /." São José do Rio Preto : [s.n.], 2010. http://hdl.handle.net/11449/94212.
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Orientador: Vanderlei Minori HoritaBanca: Daniel Smania Brandão
Banca: Ali Messaoudi
Resumo: Este trablaho apresenta os conceitos de Lyapounov e de espaços próprios e fornece um resultado devido a Oseledets, o qual trata da existência desses expoentes (e, consequentemente, dos espaços próprios) do ponto de vista da teoria da medida. A prova do teorema que nós fornecemos foi dada originalmente por Mañe e posteriormente melhorada por Viana.
Abstract: This work presents the concepts of Lyapounov exponents and of proper spaces and provides a result due to Oseledets, wich deals with the existence of these exponents (and consequently, of the proper spaces) from a measure-theoretical point of view. The proof of the theorem which we provide was originally given by Mañe later improved by Viana.
Mestre
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Chousionis, Vasileios. "Thermodynamical Formalism." Thesis, University of North Texas, 2004. https://digital.library.unt.edu/ark:/67531/metadc4631/.
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Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
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Gomes, João Tiago Assunção 1986. "Formalismos Gibbsianos para sistemas de spins unidimensionais." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307025.
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Orientador: Eduardo GaribaldiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Exibir os estados de Gibbs e os estados de equilíbrios para certos sistemas de spins sobre reticulados é um problema de grande interesse para mecânica estatística. Com este intuito, apresentamos para o caso unidimensional dois formalismos existentes para tais sistemas: o formalismo DLR (enfoque mecânico-estatístico) e o formalismo SRB (enfoque dinamicista). Apesar das particularidades próprias aos contextos nos quais cada um dos formalismos se aplica, investigam-se aqui as relações existentes entre estes através da energia livre de Gibbs e da pressão topológica. Discute-se também o comportamento assintótico dos estados de Gibbs/equilíbrio quando levados ao congelamento do sistema. Tal fenômeno nos conduz ao estudo dos estados maximizantes via teoria de otimização ergódica. Ao fim, comparam-se algumas ideias da álgebra max/min-plus e o conceito de subação, as quais serão fundamentais para análise do comportamento assintótico da pressão topológica
Abstract: To exhibit Gibbs states and equilibrium states for certain kind of lattice spin systems is a problem with great interest for statistical mechanics. To that end, we introduce two existing formalisms for one-dimensional systems: DLR formalism (statistical-mechanical approach) and SRB formalism (dynamical-systems approach). In spite of their distinct applications, we analyse the relation between them through the notions of Gibbs free energy and topological pressure. We discuss also the asymptotic behaviour of Gibbs/equilibrium states when the system is frozen. This phenomenon leads us to the study of maximizing states in the context of ergodic optimization. Finally, we compare some ideas of max/min-plus algebra and the notion of sub-action, which will be essential to investigate the asymptotic behaviour of the topological pressure
Mestrado
Matematica
Mestre em Matemática
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Mance, Bill. "Normal Numbers with Respect to the Cantor Series Expansion." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1274431587.
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