We model hyperbolic toral automorphisms by two types of Smale spaces; shifts of finite type and substitution tilings spaces. Smale spaces are dynamical systems with local hyperbolic product structure. In 1970, Bowen showed that an irreducible Smale space is a factor of a shift of finite type by showing that it has Markov partitions. Putnam extended Bowen's theorem by showing that every irreducible Smale space has a factor map that can be split into a s-bijective and u-bijective map; thereby better modelling a Smale space on its characterizing expanding and contracting spaces separately. In this thesis, we define two new constructions of Markov partitions for hyperbolic toral automorphisms inspired by the work of Adler, Weiss, and Praggastis. With one of the constructions, we investigate when a factor map from a shift of finite type to a hyperbolic toral automorphism can be written as a composition of a s-bijective and u-bijective map and we show that if such a splitting exists then the Markov partition must satisfy a Border Continuity condition. The second construction can be thought of as an explicit example of Putnam's theorem for the case of hyperbolic toral automorphisms whose defining matrix is in dimension 2 and has positive entries. We define a full splitting for all such hyperbolic toral automorphisms with one exception; the Arnold Cat map. / Graduate

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20158号 / 理博第4243号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24395号 / 理博第4894号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 並河 良典, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

<pre>In the first part of this thesis we show that, for a given non-arithmetic closed hyperbolic <i>$</i><i>n</i><i>$</i> manifold <i>$</i><i>M</i><i>$</i>, there exist for each positive integer <i>$</i><i>j</i><i>$</i>, a set <i>$</i><i>M_</i><i>1</i><i>,...,M_j</i><i>$</i> of pairwise nonisometric, strongly isospectral, finite covers of <i>$</i><i>M</i><i>$</i>, and such that for each <i>$</i><i>i,i'</i><i>$</i> one has isomorphisms of cohomology groups <i>$</i><i>H^*(M_i,</i><i>\Zbb</i><i>)=H^*(M_{i'},</i><i>\Zbb</i><i>)</i><i>$</i> which are compatible with respect to the natural maps induced by the cover. In the second part, we prove that hyperbolic <i>$</i><i>2</i><i>$</i>- and <i>$</i><i>3</i><i>$</i>-manifolds which arise from principal congruence subgroups of a maixmal order in a quaternion algebra having type number <i>$</i><i>1</i><i>$</i> are absolutely spectrally rigid. One consequence of this is a partial answer to an outstanding question of Alan Reid, concerning the spectral rigidity of Hurwitz surfaces.</pre>

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<pre>The degree of freedom brought about by breaking the directional symmetry of space through the use of anisotropic media finds applications in numerous photonic systems. Almost all these systems are based on physical principles that are generalized extensions of their isotropic counterparts, much in the same way an ellipse is related to a circle. However, as we show, there are examples where, in the presence of loss, disorder or even coupling to the measurement apparatus, emerges a completely new behavior which is qualitatively different from the isotropic case. In this work we study these emergent phenomena found in open anisotropic photonic systems.</pre> <pre><br></pre> <pre>We demonstrate that open systems based on biaxial anisotropic medium can support exceptional points which are singularities in the parameter space of the system where the mode frequencies as well as the modes themselves coalesce. We also show that topological insulators, which are novel materials that behave as dielectric in the bulk but metallic in the surface and exhibit bianisotropy through the coupling of their electric and magnetic response, can emit thermal radiation that carries nonzero spin angular momentum. Next, after describing how the strong anisotropy of hyperbolic metamaterial can support electromagnetic fields propagating with high wavenumbers unbounded by the frequency, we show that a super-resolution imaging scheme based on such material is quite robust against substantial loss and disorder. Finally, we consider an example of an incoherent perfect absorber and show that loss and anisotropy in this case can work together to recover the ideal lossless limit for the absorbing performance. In addition to making new conceptual connections between photonics and other branches of science such as condensed matter physics, biotechnology and quantum mechanics, these new emergent phenomena are shown to have thermal, imaging and sensing applications.</pre>

We begin by studying various topological properties of invariant sets of hyperbolic toral automorphisms in the linear case. Results related to cardinality, local maximality, entropy, and dimension are presented. Where possible, we extend the results to the case of hyperbolic toral automorphisms in higher dimensions, and further to general Anosov maps.

We study a system of semilinear hyperbolic PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model of SSH type of Hadad, Vitelli and Alú in [1]. We classify the system’s traveling waves, and study their stability properties. We focus on pulse solutions (solitons) on a nontrivial background and moving domain wall solutions (kinks and antikinks), corresponding to heteroclinic orbits for a reduced two-dimensional dynamical system. We further present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and the spectral stability of moving domain walls. Our result for nonlinear stability is expressed in terms of appropriately weighted ?1-norms of the perturbation, which captures the phenomenon of convective stabilization; as time advances, the traveling wave “outruns” the growing disturbance excited by an initial perturbation. We use our analytical results to interpret phenomena observed in numerical simulations. The results for (linear) spectral stability are studied in appropriately-weighted ?2-norms.

As of Russia's invasion of Ukraine we are now living in a time of full-scale war in Europe. This has ended a long period of relative peace and radically changed the security situation around the world. At the same time, the Swedish payment market has undergone a major shift in regards to cash versus digital payment. Today in Sweden, digital payments account for 96 percent of all payments, and these are made through private market actors. In response to the digitalization of the payment market, cash has been largely marginalized. Due to this coexistence, with a changed payment market and current security situation, this study aims to examine the risk management associated with payments during a potential invasion of Sweden, and the role of cash as a method of payment in wartime.  A fundamental principle assumed in this study is that the risks associated with payments in wartime are of the type Low Probability/High Consequence (LP/HC). These types of risks are particularly difficult to manage and there is thus reason to consider that the risk management of payments during an invasion may be insufficient. In this study we have constructed our own model, in order to analyze this issue, which includes three factors. These are lack of rationality, the public goods problem and hyperbolic discounting. The model aims to explain different problems and behaviors of decision-makers which may lead to inadequate risk management of LP/HC risks and thus deficient overall risk management of payments during an invasion. Throughout the study, we use the minimax-principle from game theory to further study the risk management of payments during an invasion as an LP/HC risk. In this qualitative study, eleven experts have been interviewed using a fictitious scenario of an invasion of Sweden. This functioned as the framework within which the experts then applied their insight to payments in war.  The results of the study show that the risk management of payments during an invasion has been overlooked and that lack of rationality, the public goods problem and hyperbolic discounting are all contributing factors to the problems we have identified. Further, the study defines that the payment system's lack of division of responsibilities and coordination, the unclear role of cash, as well as the cost issue and political game, are the determining factors regarding the lack of risk management around payments during an invasion. In conclusion, it is shown herein that cash does have a role to play during an invasion, but that extensive reformations must be made to increase the infrastructure around cash management.

In this thesis, we address several problems related to the bosonic classical and algebraic quantum field theories in curved spacetime. In particular, the main question is: how do the theories change under finite global variations of the spacetime metric tensor? To answer this question a new deformation tool, the paracausal deformation, is developed and studied on its own as a new approach to investigate the structure of the space of globally hyperbolic metric tensors associated with a smooth manifold $\M$. Then the classical M{\o}ller maps are constructed to compare solutions of the hyperbolic PDEs defining the classical field theories and the quantum M{\o}ller $*$-isomorphisms follow to compare the CCR quantum algebras associated to the propagation of the quantum fields on the different background geometries. These maps possess the important property of preserving Hadamard states, providing a new way to implement the deformation argument used to prove the existence of such states in general globally hyperbolic spacetime. Moreover, the algebraic quantization of the Proca field, i.e the massive spin 1 field, on a general globally hyperbolic spacetime is for the first time studied in detail: by employing techniques coming from microlocal analysis and spectral theory a Hadamard state is constructed on ultrastatic spacetimes and then the M{\o}ller operator is used to prove the existence of such states in general globally hyperbolic spacetimes. A discussion about the definition of Hadamard states for the massive vector fields closes the work. The thesis is based on two works on algebraic quantization of bosonic field theories and Hadamard states: \cite{Norm}, \cite{Proca}. The papers are co-authored by my supervisor Prof. Valter Moretti (UniTN) and cosupervisor Simone Murro (UniGe). The first \cite{DefArg1} has not been included since, at the time it was written, the paracausal deformation, the construction of M{\o}ller operators, the right approach to intertwine the causal propagators and all the other tools developed in the subsequent works were still at a rough stage.