Partial hyperbolicity and attracting regions in 3-dimensional manifolds

En esta tesis se estudia por un lado la existencia, unicidad y estructura geométrica de atractores y quasi-atractores, y por otro la estructura geométrica de las foliaciones invariantes para ciertos sistemas parcialmente hiperbólicos. El estudio requiere entrar en el estudio sistemático de ciertas foliaciones que aparecen y permite probar la integrabilidad de ciertos fibrados invariantes. El trabajo tiene apéndices presentando trabajos relacionados aunque no directamente.Agencia Nacional de Investigación e Innovació

A Cognitive Information Theory of Music: A Computational Memetics Approach

This thesis offers an account of music cognition based on information theory and memetics. My research strategy is to split the memetic modelling into four layers: Data, Information, Psychology and Application. Multiple cognitive models are proposed for the Information and Psychology layers, and the MDL best-fit models with published human data are selected. Then, for the Psychology layer only, new experiments are conducted to validate the best-fit models. In the information chapter, an information-theoretic model of musical memory is proposed, along with two competing models. The proposed model exhibited a better fit with human data than the competing models. Higher-level psychological theories are then built on top of this information layer. In the similarity chapter, I proposed three competing models of musical similarity, and conducted a new experiment to validate the best-fit model. In the fitness chapter, I again proposed three competing models of musical fitness, and conducted a new experiment to validate the best-fit model. In both cases, the correlations with human data are statistically significant. All in all, my research has shown that the memetic strategy is sound, and the modelling results are encouraging. Implications of this research are discussed

On Conley's fundamental theorem of dynamical systems

We generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition

The fundamental theorem of dynamical systems

summary:We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem should be called The Fundamental Theorem of Dynamical Systems

A Conley-type Lyapunov function for the strong chain recurrent set

Let $\phi:X\times\mathbb{R} \rightarrow X$ be a continuous flow on a compact metric space $(X,d)$. In this article we constructively prove the existence of a continuous Lyapunov function for $\phi$ which is strictly decreasing outside $\mathcal{SCR}_d(\phi)$. Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set