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In this dissertation, matching, entropy, holes and expansions come together. The first chapter is an introduction to ergodic theory and dynamical systems. The second chapter is on, what we called Flipped $\alpha$-expansions. For this family we have an invariant measure that is $\sigma$-finite infinite. We calculate the Krengel entropy for a large part of the parameter space and find an explicit expression for the density by using the natural extension. In Chapter 3 Ito Tanaka's $\alpha$-continued fractions are studied. We prove that matching holds almost everywhere and that the non-matching set has full Hausdorff dimension. In the fourth chapter we study $N$-expansions with flips. We use a Gauss-Kuzmin-Levy method to approximate the density for a large family and use this to give an estimation for the entropy. In the last Chapter we look at greedy $\beta$-expansions. We show that for almost every $\beta\in(1,2]$ the set of points $t$ for which the forward orbit avoids the hole $[0,t)$ has infinitely many isolated and infinitely many accumulation points in any neighborhood of zero. Furthermore, we characterize the set of $\beta$ for which there are no accumulation points and show that this set has Hausdorff dimension zero.

This work presents a new graphical system to represent hydrological dynamical models and their interactions. We propose an extended version of the Petri Nets mathematical modeling language, the Extended Petri Nets (EPN), which allows for an immediate translation from the graphics of the model to its mathematical representation in a clear way. We introduce the principal objects of the EPN representation (i.e., places, transitions, arcs, controllers, and splitters) and their use in hydrological systems. We show how to cast hydrological models in EPN and how to complete their mathematical description using a dictionary for the symbols and an expression table for the flux equations. Thanks to the compositional property of EPN, we show how it is possible to represent either a single hydrological response unit or a complex catchment where multiple systems of equations are solved simultaneously. Finally, EPN can be used to describe complex Earth system models that include feedback between the water, energy, and carbon budgets. The representation of hydrological dynamical systems with EPN provides a clear visualization of the relations and feedback between subsystems, which can be studied with techniques introduced in nonlinear systems theory and control theory.

Earthquakes, as a natural phenomenon and their consequences upon structures, have been addressed from deterministic, pseudo-empirical and primary statistical-probabilistic points of view. In the latter approach, 'primary' is meant to suggest that randomness has been artificially introduced into the variables of investigation. An alternative view has been advanced by a number ofresearchers that have classified earthquakes as chaotic from an ontological perspective. Their arguments are founded in the high degree of non-linearity of the equations ruling the corresponding seismic waves. However, the sensitivity of long time behavior of dynamic systems to variations in initial conditions, known as the Chaos Paradigm appears as a by-product of a deeper insight into natural phenomena known as Theory of Open Dynamical Systems ODS. An open system is currently defined as the relation between a part of nature, the main system which contains the observations we make, and its surrounding environment. ODS theory has been applied to different research subjects including physics, chemistry, and biology, for identifying and controlling undesired chaotic behavior in highly nonlinear dynamic systems. It is suggested that earthquakes and their interaction with structures constitute an example of an open system. Recognizing that in Earthquake Engineering the application of those concepts has not been previously investigated, in this paper a discussion related to the use of ODS concepts in that particular field is presented. Using the most basic case of a linear elastic single degree offreedom SDOF oscillator, differences in the prediction of the response of the system subjected to only one ground motion using a Newton classical approach and ODS concepts, which involve stochastic processes, are compared. Conclusions about the consequences of the application of ODS theory for re-understanding Earthquake Engineering are presented, and a general critique to primary probabilistic approaches for addressing the same problem is formulated.

Earthquakes, as a natural phenomenon and their consequences upon structures, have been addressed from deterministic, pseudo-empirical and primary statistical-probabilistic points of view. In the latter approach, 'primary' is meant to suggest that randomness has been artificially introduced into the variables of investigation. An alternative view has been advanced by a number ofresearchers that have classified earthquakes as chaotic from an ontological perspective. Their arguments are founded in the high degree of non-linearity of the equations ruling the corresponding seismic waves. However, the sensitivity of long time behavior of dynamic systems to variations in initial conditions, known as the Chaos Paradigm appears as a by-product of a deeper insight into natural phenomena known as Theory of Open Dynamical Systems ODS. An open system is currently defined as the relation between a part of nature, the main system which contains the observations we make, and its surrounding environment. ODS theory has been applied to different research subjects including physics, chemistry, and biology, for identifying and controlling undesired chaotic behavior in highly nonlinear dynamic systems. It is suggested that earthquakes and their interaction with structures constitute an example of an open system. Recognizing that in Earthquake Engineering the application of those concepts has not been previously investigated, in this paper a discussion related to the use of ODS concepts in that particular field is presented. Using the most basic case of a linear elastic single degree offreedom SDOF oscillator, differences in the prediction of the response of the system subjected to only one ground motion using a Newton classical approach and ODS concepts, which involve stochastic processes, are compared. Conclusions about the consequences of the application of ODS theory for re-understanding Earthquake Engineering are presented, and a general critique to primary probabilistic approaches for addressing the same problem is formulated. Los terremotos, como fenómeno natural y sus consecuencias en estructuras han sido abordados desde perspectivas deterministas, pseudo-empíricas y desde un punto de vista probabilístico-estadístico primario. En este último enfoque, el adjetivo 'primario' significa que la aleatoriedad ha sido introducida en forma artificial en las variables de investigación relevantes en Ingeniería Sísmica. Una perspectiva alternativa ha sido planteada por investigadores que han clasificado los terremotos, desde un punto de vista ontológico, como fenómenos caóticos. Sus argumentos se fundamentan en el alto grado de no-linealidad y complejidad de las ecuaciones que gobiernan la ecuación de las ondas sísmicas. Sin embargo, la sensibilidad de la respuesta dinámica en el largo plazo debido a pequeñas variaciones en las condiciones iniciales, conocido como el Paradigma del Caos, aparece como un subproducto de una mirada más profunda dentro de fenómenos naturales en general, llamada Teoría de Sistemas Dinámicos Abiertos ODS. Un sistema abierto se define como la relación entre una parte de la naturaleza, el sistema primario que contiene nuestras observaciones, y el sistema que lo contiene. Los principios de ODS han sido aplicados a distintas disciplinas científicas tales como la física, la química y la biología, con la intensión de identificar y controlar comportamientos caóticos no deseados en sistemas altamente no-lineales. Se postula que los terremotos y su interacción con estructuras constituyen un caso de un sistema abierto. Reconociendo que en Ingeniería Sísmica la aplicación de dichos conceptos no ha sido anteriormente investigada, en este artículo se presenta una discusión sobre su posible uso en dicha disciplina. Utilizando un modelo que representa el caso más básico de un oscilador de un grado de libertad con rigidez lineal elástica sometido sólo a un registro sísmico, se discuten las diferencias en la respuesta del oscilador obtenida usando la dinámica clásica de Newton y conceptos de ODS que incluyen procesos estocásticos como el que se utiliza en esta contribución. Conclusiones sobre las consecuencias de la aplicación de ODS son formuladas para re-entender la Ingeniería Sísmica, incorporando una crítica a enfoques probabilísticos primarios para abordar el mismo problema.

Dynamical systems that are close to non-ergodic are characterised by the existence of subdomains or regions whose trajectories remain confined for long periods of time. A well-known technique for detecting such metastable subdomains is by considering eigenfunctions corresponding to large real eigenvalues of the Perron-Frobenius transfer operator. The focus of this thesis is to investigate the asymptotic behaviour of trajectories exiting regions obtained using such techniques. We regard the complement of the metastable region to be a ‘hole’, and show in Chapter 2 that an upper bound on the escape rate into the hole is determined by the corresponding eigenvalue of the Perron-Frobenius operator. The results are illustrated via examples by showing applications to uniformly expanding maps of the unit interval. In Chapter 3 we investigate a non-uniformly expanding map of the interval to show the existence of a conditionally invariant measure, and determine asymptotic behaviour of the corresponding escape rate. Furthermore, perturbing the map slightly in the slowly expanding region creates a spectral gap. This is often observed numerically when approximating the operator with schemes such as Ulam’s method. We investigate the asymptotic scaling of the spectral gap as the perturbation vanishes. In Chapter 4 we consider escape rate from random sets under the action of random dynamics and prove a result analogous to that of Chapter 2. We also show, under fairly weak assumptions, that in Oseledets subspaces Lyapunov exponents with respect to different norms are equal. The results are applied to Rychlik random dynamical systems. Finally, Chapter 5 deals with the main themes of the earlier chapters in the settings of deterministic and random shifts of finite type. There, we demonstrate methods to decompose shifts into complementary subshifts of large entropy. Much of the material in this thesis has either appeared in a scientific journal or has been submitted to one for publication.