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
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
