We study the hyperbolic dynamical systems known as Smale spaces. More specifically we investigate the C*-algebras constructed from these systems. The K group of one of these algebras has a natural ring structure arising from an asymptotically abelian property. The K groups of the other algebras are then modules over this ring. In the case of a shift of finite type we compute these structures explicitly and show that the stable and unstable algebras exhibit a certain type of duality as modules. We also investigate the Bowen measure and its stable and unstable components with respect to resolving factor maps, and prove several results about the traces that arise as integration against these measures. Specifically we show that the trace is a ring/module homomorphism into R and prove a result relating these integration traces to an asymptotic of the usual trace of an operator on a Hilbert space.

Une variété riemannienne est dite rigide lorsque la longueur des géodésiques périodiques (cas des variétés fermées) ou des géodésiques diffusées (cas des variétés ouvertes) permet de reconstruire globalement la géométrie de la variété. Cette notion trouve naturellement son origine dans des dispositifs d’imagerie numérique tels que la tomographie par rayons X. Grâce une approche résolument analytique initiée par Guillarmou et fondée sur de l’analyse microlocale (plus particulièrement sur certaines techniques récentes dues à Faure-Sjostrand et Dyatlov-Zworski permettant une étude analytique fine des flots Anosov), nous montrons que le spectre marqué des longueurs, c’est-à-dire la donnée des longueurs des géodésiques périodiques marquées par l’homotopie, d’une variété fermée Anosov ou Anosov à pointes hyperboliques détermine localement la métrique de la variété. Dans le cas d’une variété ouverte avec ensemble capté hyperbolique, nous montrons que la distance marquée au bord, c’est-à-dire la donnée de la longueur des géodésiques diffusées marquées par l’homotopie, détermine localement la métrique. Enfin, dans le cas d’une surface asymptotiquement hyperbolique, nous montrons qu’une notion de distance renormalisée entre paire de points au bord à l’infini permet de reconstruire globalement la géométrie de la surface. / A Riemannian manifold is said to be rigid if the length of periodic geodesics (in the case of a closed manifold) or scattered geodesics (in the case of an open manifold) allows to recover the full geometry of the manifold. This notion naturally arises in imaging devices such as X-ray tomography. Thanks to a analytic framework introduced by Guillarmou and based on microlocal analysis (and more precisely on the analytic study of hyperbolic flows of Faure-Sjostrand and Dyatlov-Zworski), we show that the marked length spectrum, that is the lengths of the periodic geodesics marked by homotopy, of a closed Anosov manifold or of an Anosov manifold with hyperbolic cusps locally determines its metric. In the case of an open manifold with hyperbolic trapped set, we show that the length of the scattered geodesics marked by homotopy locally determines the metric. Eventually, in the case of an asymptotically hyperbolic surface, we show that a suitable notion of renormalized distance between pair of points on the boundary at infinity allows to globally reconstruct the geometry of the surface.

No description available.

Markov partitions provide a 'good' mechanism of symbolic dynamics for uniformly hyperbolic systems, forming the classical foundation for the thermodynamic formalism in this setting, and remaining useful in the modern theory. Usually, however, one takes Bowen's 1970's general construction for granted, or restricts to cases with simpler geometry (as on surfaces) or more algebraic structure. This thesis examines several questions on the algorithmic content of (topological) Markov partitions, starting with the pointwise, entropy-like, topological conjugacy invariant known as orbit complexity. The relation between orbit complexity de nitions of Brudno and Galatolo is examined in general compact spaces, and used in Theorem 2.0.9 to bound the decrease in some of these quantities under semiconjugacy. A corollary, and a pointwise analogue of facts about metric entropy, is that any Markov partition produces symbolic dynamics matching the original orbit complexity at each point. A Lebesgue-typical value for orbit complexity near a hyperbolic attractor is also established (with some use of Brin-Katok local entropy), and is technically distinct from typicality statements discussed by Galatolo, Bonanno and their co-authors. Both our results are proved adapting classical arguments of Bowen for entropy. Chapters 3 and onwards consider the axiomatisation and computable construction of Markov partitions. We propose a framework of 'abstract local product structures'

In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available. / I den här uppsatsen studerar vi två olika men relaterede treparameterfamiljer av kontinuerligt differentierbara avbildningar från enhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en mängd av positivt mått av parametrar sådana att det finns en mängd av positivt mått så att för varje punkt i den senarenämnde mängden erfar derivatan exponentiell tillväxt. Vi uppnår detta genom att använda en metod som liknar den som Michael Benedicks och Lennart Carleson använde för att studera den kvadratiska familjen. För den andra familjen försöker vi visa ett liknande men svagare resultat genom att använda en liknande metodik men misslyckas. Vi diskuterar och förklarar vilka svårigheter den senare familjen ger och vilka egenskaper som Benedicks och Carleson använder sig av hos den kvadratiska familjen som vår familj saknar

In the present work, we introduce a never before studied soliton equation called the intermediate mixed Manakov (IMM) equation. Through a pole ansatz, we prove that the equation has N-soliton solutions with pole parameters governed by the hyperbolic Calogero-Moser system. We also show that there are spatially periodic N-soliton solutions with poles obeying elliptic Calogero-Moser dynamics. A Lax pair is given in the form of a Riemann-Hilbert problem on a cylinder. A similar Lax pair is shown to imply a novel spin generalization of the intermediate nonlinear Schrödinger equation. Some conservation laws for the IMM are proven. We demonstrate that the IMM can be written as a Hamiltonian system, with one of these conserved quantities as the Hamiltonian. Finally, a physical interpretation is given by showing that the IMM can be rewritten to describe a system of two nonlocally coupled fluids, with nonlinear self-interactions. / Vi presenterar en aldrig tidigare studerad solitonekvation som vi döper till ‘the intermediate mixed Manakov equation’ (ungefär ‘den mellanliggande kopplade Manakovekvationen’. Kortform: IMM). Genom en polansats bevisar vi att ekvationen har N-solitonlösningar där polparametrarna utgör ett hyperboliskt Calogero-Mosersystem. Vi visar också att det finns rumsligt periodiska N-solitonlösningar vars poler följer elliptisk Calogero-Moserdynamik. Ett Laxpar ges i form av ett Riemann-Hilbertproblem på en cylinder. Vi demonstrerar att ett liknande Laxpar leder till en ny spinngeneralisering av den s.k. INLS-ekvationen. Några bevarandelagar för IMM bevisas. Vi visar att IMM-ekvationen kan skrivas som ett Hamiltonskt system, där Hamiltonianen är en av våra tidigare bevarade storheter. Till sist ger vi en fysikalisk tolkning av vår ekvation genom att demonstrera hur den beskriver ett system av ickelokalt interagerande vätskor, med ickelinjära självinteraktioner.