This bachelor's thesis discusses from an ergodic-theoretical perspective the "metric functional analysis" that Anders Karlsson and others have developed in the recent years. We introduce a new symbolic calculus for metric functionals which includes a notion of the adjoint of a nonexpansive map. Using these tools we revisit many central results, including Karlsson's spectral principle and its stronger form for star-shaped spaces due to Gaubert and Vigeral, as well as the multiplicative ergodic theorem of Karlsson-Ledrappier.

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

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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.

[pt] Apresentaremos uma versão de Teorema Ergódico Multiplicativo para cociclos subaditivos devido a Karlsson e Margulis. Como aplicação, analisaremos três exemplos de cociclos nos seguintes espaços: Grafo gerado por grupo livre em dois geradores, disco hiperbólico, espaco das matrizes positivas simétricas definidas. Também usaremos o Teorema de Karlsson e Margulis para mostrar o Teorema de Oseledets. / [en] We will show a version of Multiplicative Ergodic Theorem for subbaditive cocycles due to Karlsson and Margulis. As an application, we will analyze three examples of cocycles in following spaces: graph generated by free group of two generators, hyperbolic disc, space of positive definite symetric matrices. Also, we will use the Theorem of Karlsson and Margulis to prove Theorem of Oseledets.

This dissertation examines the role of the spatial in four works of digital interactive literature. These works are Dreamaphage by Jason Nelson (2003), Last Meal Requested by Sachiko Hayashi (2003), Façade by Michael Mateas and Andrew Stern (2005) and Egypt: The Book of Going Forth by Day by M. D. Coverley (2006). The study employs an original analytical method based on close reading and spatial analysis, which combines narrative, design and interaction theories. The resulting critique argues that the spatial components of the digital works define reader interaction and the narratives that result from it. This is one of very few in-depth studies grounded in the close reading of the spatial in digital interactive literature. Over five chapters, the dissertation analyzes the four digital works according to three common areas. Firstly, the prefaces, design and addressivity are present in each. Secondly, each of the works relies on the spatial for both interaction and the meanings that result. Thirdly, the anticipation of responses from a reader is evaluated within the interactive properties of each work. This anticipation is coordinated across the written text, moving and still images, representations of places, characters, audio and navigable spaces. The similar divisions of form, the role of the spatial and the anticipation of responses provide the basic structure for analysis. As a result, the analytical chapters open with an investigation of the prefaces, move on to the design and conclude with how the spaces of the digital works can be addressive or anticipate responses. In each chapter representations of space and representational space are described in relation to the influence they have upon the potentials for reader interaction as spatial practice. This interaction includes interpretation, as well as those elements associated with the ergodic, or the effort that defines the reception of the digital interactive texts. The opening chapter sets out the relevant theory related to space, interaction and narrative in digital literature. Chapter two presents the methodology for close reading the spatial components of the digital texts in relation to their role in interaction and narrative development. Chapter three assesses the prefaces as paratextual thresholds to the digital works and how they set up the spaces for reader engagement. The next chapter takes up the design of the digital works and its part in the formation of space and how this controls interaction. The fifth chapter looks at the addressivity of the spatial and how it contributes to the possibilities for interaction and narrative. The dissertation argues for the dominance of the spatial as a factor within the formation of narrative through interaction in digital literature, with implications across contemporary storytelling and narrative theory.

We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.