This dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are... Show moreThis dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are critically intermittent. In Chapter 2 we consider a large class of such systems and demonstrate the existence of a phase transition, where the absolutely continuous invariant measure changes between finite and infinite. For a closely related class we derive in Chapter 3 statistical properties like decay of correlations and the Central Limit Theorem. In Chapter 4 we investigate whether a similar phase transition remains to exist when the critical behaviour is toned down. Random interval maps can also be used to generate number expansions, which will be the main object of study in the second part. In Chapter 5 we generalize Lochs’ Theorem, which compares the efficiency between representing real numbers in decimal expansions and regular continued fraction expansions, to a wide class of pairs of random interval maps that produce number expansions. Closely related to this result, we study in Chapter 6 the efficiency of beta-encoders as a potential source for pseudo-random number generation by comparing the output of a beta-encoder with its corresponding binary expansion. Show less
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... Show moreIn 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. Show less
In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using... Show moreIn this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin–Lévy based approximation method is used. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N-expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N=2 and N=36. Interesting behavior can be observed from numerical results. Show less
Central to this research is the Urban Future-project, which consists of a large archive of artworks made from 2002 until now. The original question underpinning this project was: what... Show moreCentral to this research is the Urban Future-project, which consists of a large archive of artworks made from 2002 until now. The original question underpinning this project was: what influence do chaos, entropy and fragmentation have on the viability of the rapidly developing urbanizing world? In the course of the research project, the (literature and field) explorations led to the assumption that there is a demonstrable and necessary link between the quality of life in the city and vital social cohesion on the one hand and chaos, entropy and fragmentation on the other. In the artistic part of the research focuses on the question: is it possible to make the supposed connection between quality of urban life and chaos, entropy and fragmentation visible in artwork and, if so, how? In the written dissertation, working methods and strategies are contextualized and analyzed. The visual part derives from an artist's position which uses non-verbal, sensorial strategies to reach new insights. It mainly focuses on the visual and aesthetic possibilities of aspects of fragmentation, chaos and entropy because Scholten considers these aspects, as productive forces, to be the core of the experience of urbanization. Show less
