In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.

This paper studies symbolic dynamical systems {0, 1}G, where G is a countably infinite group, {0, 1}G has the product topology, and G acts on {0, 1}G by shifts. It is proven that for every countably infinite group G the union of the minimal free subflows of {0, 1}G is dense. In fact, a stronger result is obtained which states that if G is a countably infinite group and U is an open subset of {0, 1}G, then there is a collection of size continuum consisting of pairwise disjoint minimal free subflows intersecting U.

Topological evolution is a new dynamical systems model of biological evolution occurring within a genomic state space. It can be modeled equivalently as a stochastic dynamical system, a stochastic differential equation, or a partial differential equation drift-diffusion model. An application of this approach is a model of disease evolution tracing diseases in ways similar to standard functional traits (e.g., organ evolution). Genetically embedded diseases become evolving functional components of species-level genomes. The competition between species-level evolution (which tends to maintain diseases) and individual evolution (which acts to eliminate them), yields a novel structural topology for the stochastic dynamics involved. In particular, an unlimited set of dynamical time scales emerges as a means of timing different levels of evolution: from individual to group to species and larger units. These scales exhibit a dynamical tension between individual and group evolutions, which are modeled on very different (fast and slow, respectively) time scales. This is analyzed in the context of a potentially major constraint on evolution: the species-level enforcement of lifespan via (topological) barriers to genomic longevity. This species-enforced behavior is analogous to certain types of evolutionary altruism, but it is denoted here as extreme altruism based on its potential shaping through mass extinctions. We give examples of biological mechanisms implementing some of the topological barriers discussed and provide mathematical models for them. This picture also introduces an explicit basis for lifespan-limiting evolutionary pressures. This involves a species-level need to maintain flux in its genome via a paced turnover of its biomass. This is necessitated by the need for phenomic characteristics to keep pace with genomic changes through evolution. Put briefly, the phenome must keep up with the genome, which occurs with an optimized limited lifespan. An important consequence of this model is a new role for diseases in evolution. Rather than their commonly recognized role as accidental side-effects, they play a central functional role in the shaping of an optimal lifespan for a species implemented through the topology of their embedding into the genome state space. This includes cancers, which are known to be embedded into the genome in complex and sometimes hair-triggered ways arising from DNA damage. Such cancers are known also to act in engineered and teleological ways that have been difficult to explain using currently very popular theories of intra-organismic cancer evolution. This alternative inter-organismic picture presents cancer evolution as occurring over much longer (evolutionary) time scales rather than very shortened organic evolutions that occur in individual cancers. This in turn may explain some evolved, intricate, and seemingly engineered properties of cancer. This dynamical evolutionary model is framed in a multiscaled picture in which different time scales are almost independently active in the evolutionary process acting on semi-independent parts of the genome. We additionally move from natural evolution to artificial implementations of evolutionary algorithms. We study genetic programming for the structured construction of machine learning features in a new structural risk minimization environment. While genetic programming in feature engineering is not new, we propose a Lagrangian optimization criterion for defining new feature sets inspired by structural risk minimization in statistical learning. We bifurcate the optimization of this Lagrangian into two exhaustive categories involving local and global search. The former is accomplished through local descent with given basins of attraction while the latter is done through a combinatorial search for new basins via an evolution algorithm.

In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.

We investigate two combinatorial properties of classes of finite structures, as well as related applications to topological dynamics. Using the Hrushovski property of classes of finite structures -- a finite extension property of homomorphisms -- we can show the existence of ample generics. For example, Solecki proved the existence of ample generics in the context of finite metric spaces that do indeed possess this extension property. Furthermore, Kechris, Pestov and Todorcevic have shown that the Ramsey property of Fraisse classes of finite structures implies that the automorphism group of the corresponding Fraisse limit is extremely amenable, i.e., it possesses a very strong fixed point property. Gromov and Milman had shown that the unitary group of the infinite-dimensional separable Hilbert space is extremely amenable using non-combinatorial methods. This result encourages a deeper look into structural Euclidean Ramsey theory, i.e., Euclidean Ramsey theory in which we colour more than just points. In particular, we look at complete finite labeled graphs whose vertex sets are subsets of the Hilbert space and whose labels correspond to the inner products. We prove "Ramsey-type" and "Hrushovski-type" theorems for linearly ordered metric subspaces of "sufficiently" orthogonal sets. In particular, the latter is used to show a "Hrushovski version" of the Ramsey-type Matousek-Rodl theorem for simplices. It is known that the square root of the metric induced by the distance between vertices in graphs produces a metric space embeddable in a Euclidean space if and only if the graph is a metric subgraph of the Cartesian product of three types of graphs. These three are the half-cube graphs, the so-called cocktail party graphs, and the Gosset graph. We show that the class of metric spaces related to half-cube graphs -- metric spaces on sets with the symmetric difference metric -- satisfies the Hrushovski property up to 3 points, but not more. Moreover, the amalgamation in this class can be too restrictive to permit the Ramsey Property. Finally, following the work of Fouche, we compute the Ramsey degrees of structures induced by the leaf sets of boron trees. Also, we briefly show that this class does not satisfy the full Hrushovski property. Fouche's trees are in fact related to ultrametric spaces, as was observed by Lionel Nguyen van The. We augment Fouche's concept of orientation so that it applies to these boron tree structures. The upper bound computation of the Ramsey degree in this case, turns out to be an "asymmetric" version of the Graham-Rothschild theorem. Finally, we extend these structures to "oriented" ones, yielding a Ramsey class and a corresponding Fraisse limit whose automorphism group is extremely amenable.

We investigate two combinatorial properties of classes of finite structures, as well as related applications to topological dynamics. Using the Hrushovski property of classes of finite structures -- a finite extension property of homomorphisms -- we can show the existence of ample generics. For example, Solecki proved the existence of ample generics in the context of finite metric spaces that do indeed possess this extension property. Furthermore, Kechris, Pestov and Todorcevic have shown that the Ramsey property of Fraisse classes of finite structures implies that the automorphism group of the corresponding Fraisse limit is extremely amenable, i.e., it possesses a very strong fixed point property. Gromov and Milman had shown that the unitary group of the infinite-dimensional separable Hilbert space is extremely amenable using non-combinatorial methods. This result encourages a deeper look into structural Euclidean Ramsey theory, i.e., Euclidean Ramsey theory in which we colour more than just points. In particular, we look at complete finite labeled graphs whose vertex sets are subsets of the Hilbert space and whose labels correspond to the inner products. We prove "Ramsey-type" and "Hrushovski-type" theorems for linearly ordered metric subspaces of "sufficiently" orthogonal sets. In particular, the latter is used to show a "Hrushovski version" of the Ramsey-type Matousek-Rodl theorem for simplices. It is known that the square root of the metric induced by the distance between vertices in graphs produces a metric space embeddable in a Euclidean space if and only if the graph is a metric subgraph of the Cartesian product of three types of graphs. These three are the half-cube graphs, the so-called cocktail party graphs, and the Gosset graph. We show that the class of metric spaces related to half-cube graphs -- metric spaces on sets with the symmetric difference metric -- satisfies the Hrushovski property up to 3 points, but not more. Moreover, the amalgamation in this class can be too restrictive to permit the Ramsey Property. Finally, following the work of Fouche, we compute the Ramsey degrees of structures induced by the leaf sets of boron trees. Also, we briefly show that this class does not satisfy the full Hrushovski property. Fouche's trees are in fact related to ultrametric spaces, as was observed by Lionel Nguyen van The. We augment Fouche's concept of orientation so that it applies to these boron tree structures. The upper bound computation of the Ramsey degree in this case, turns out to be an "asymmetric" version of the Graham-Rothschild theorem. Finally, we extend these structures to "oriented" ones, yielding a Ramsey class and a corresponding Fraisse limit whose automorphism group is extremely amenable.

No description available.

On s'intéresse dans ce travail aux automates cellulaires unidimensionnels qui ont été largement étudiés mais où il reste beaucoup à faire. La théorie spectrale des automates cellulaires a notamment été peu abordée à l'exception de quelques résultats indirects. On cherche a mieux comprendre les cadres topologiques et ergodiques en étudiant l'existence de valeurs propres en particulier celles irrationnelles c'est à dire de la forme e^{2Iπα} où α est un irrationnel et I la racine carrée de l'unité. Cette question ne semble pas avoir été abordée jusqu'à présent. Dans le cadre topologique les résultats sur l'équicontinuité de Kůrka et Blanchard et Tisseur permettent de déduire directement que tout automate cellulaire équicontinu possède des valeurs propres topologiques rationnelles. La densité des points périodiques pour le décalage empêche l'existence de valeurs propres topologiques irrationnelles. La densité des points périodiques pour l'automate cellulaire semble être liée à la question des valeurs propres. Dans le cadre topologique, si l'automate cellulaire possède des points d'équicontinuité sans être équicontinu, la densité des points périodiques a comme conséquence le fait que le spectre représente l'ensemble des racines rationnelles de l'unité c'est à dire tous les nombres de la forme e^{2Iπα} avec α∈Q .Dans le cadre mesuré, la question devient plus difficile, on s'intéresse à la dynamique des automates cellulaires surjectifs pour lesquels la mesure uniforme est invariante en vertu du théorème de Hedlund. La plupart des résultats obtenus demeurent valable dans un cadre plus large. Nous commençons par montrer que les automates cellulaires ayant des points d'équicontinuité ne possèdent pas de valeurs propres mesurables irrationnelles. Ce résultat se généralise aux automates cellulaires possédant des points μ-équicontinu selon la définition de Gilman. Nous démontrons finalement que les automates cellulaires possédant des points μ-équicontinu selon la définition de Gilman possèdent des valeurs propres rationnelles / We investigate properties of one-dimensional cellular automata. This category of cellular automata has been widely studied but many questions are still open. Among them the spectral theory of unidimensional cellular automata is an open field with few indirect results. We want a better understanding of both ergodic and topological aspect by investigating the existence of eigenvalues of cellular automata, in particular irrational ones, i.e., those of the form e^{2Iπα} where α is irrationnal and I the complex root of -1. The last question seems not to have been studied yet.In the topological field the results of Kůrka & Blanchard and Tisseur about equicontinuous cellular automata have as direct consequence that any equicontinuous CA has rational eigenvalues. Density of shift periodic points leads to the impossibility for CA to have topological irrational eigenvalues. The density of periodic points of cellular automata seems to be related with the question of eignevalues. If the CA has equicontinuity points without being equicontinuous, the density of periodic points implies the fact that the spectrum contains all rational roots of the unity, i.e., all numbers of the form e^{2Iπα} with α∈Q .In the measurable field the question becomes harder. We assume that the cellular automaton is surjective, which implies that the uniform measure is invariant. Most results are still available in more general conditions. We first prove that cellular automata with equicontinuity points never have irrational measurable eigenvalues. This result is then generalized to cellular automata with μ-equicontinuous points according to Gilman's classification. We also prove that cellular automata with μ-equicontinuous points have rational eigenvalues