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
Charlene, C.C.C.J.; Langeveld, N.D.S.; Maggioni, M.; Munday, S. 2020
As a natural counterpart to Nakada's α-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds... Show moreAs a natural counterpart to Nakada's α-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite σ-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps. Show less
