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
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
