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
This thesis deals with Markov operators and semigroups. A Markov operator is a positive linear operator on the space of finite measures on some state space that preserves mass. A Markov semigroup... Show moreThis thesis deals with Markov operators and semigroups. A Markov operator is a positive linear operator on the space of finite measures on some state space that preserves mass. A Markov semigroup is a family of Markov operators parametrised by the positive real numbers, satisfying the semigroup property. These appear naturally in various places: deterministic dynamical systems, iterated function systems, structured population models and more generally Markov chains and Markov processes. We will study general Markov operators and semigroups in a functional analytic framework. Because the usual topology on the space of measures, given by the total variation norm, is often too strong for applications, we consider weaker topologies on the space of measures. and study continuity properties of Markov semigroups and their restriction to invariant subspaces. In the latter part of the thesis we provide ergodic decompositions, yielding, among other things, a characterisation of ergodic measures and an 'explicit' integral decomposition of invariant measures into ergodic measures. Under extra equicontinuity assumptions the ergodic decompositions have some nice properties, allowing us to find various characterisations for the existence, uniqueness, mean ergodicity and stability of invariant measures, and giving us extra information on the set of ergodic measures. Show less
