This thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure... Show moreThis thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure transformations, not necessarily number systems. Natural extensions, the Perron-Frobeniusoperator and the dynamical phenomenon of matching are some of the techniques exploited to obtain such results. In particular, in this thesis the notion of matching is for the first time recognised in an infinite measure system and the definition, known so far for deterministic transformations only, is extended to cover random interval maps as well. The thesis also presents new developments in the area of number expansions 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
This PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the... Show moreThis PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the classical Hilbert space setting to the setting of general CAT(0) spaces. In the third chapter, the non-symmetric Fokker-Planck equation is treated as a flow on the Wasserstein-2 space of probability measures, and it is proven that its semigroup of solutions possesses similar properties to those of the gradient flow semigroups. In the forth chapter, a general theory of maximal monotone operators and the induced flows on Wasserstein-2 spaces is developed. This theory generalizes the theory of gradient flows by Ambrosio-Gigli-Savare. In the final fifth chapter the problem of the existence of an invariant measure for stochastic delay equations is proven. The diffusion coefficient has delay, and is assumed to be locally Lipschitz and bounded. Show less
