On the Dimension Spectrum of Infinite Subsystems of Continued Fractions and Perron-Frobenius Operators

The main topic of this thesis centres around the dimension spectrum of continued fractions expansions and the use of the spectral theory of Perron-Frobenius operators in this context. The study of the dimension spectrum of iterated function systems has a long history going back to the famous Texan conjecture, which asserts that the dimension spectrum of the iterated function system Θ ={θn: n ∈ N}, where θn(x) = 1 n+x , is full, i.e., for each 0 ≤ s ≤ 1, there is an F ⊆N such that dimH(JF) = s. Here JF is the collection of all irrationals x ∈(0,1) whose continued expansion digits belong to F, so

JF ={x=[a1,a2,a3,...]: ai ∈ F,i ∈ N} ⊆ [0,1].

This conjecture was settled by Kesseböhmer and Zhu in [30]. Later, other interesting iterated function systems were shown to have a full dimension spectrum, including, reverse continued fractions expansions [18] and complex continued fractions expansions [8]. Since then, the structure of the dimension spectrum DS(Θ) has been studied for various classes of iterated function systems, see [7, 8, 10, 19, 29]. Among other results it was shown [8] that, the dimension spectrum of a conformal iterated function system is a compact and perfect set.

Recently the dimension spectrum of infinite subsystems ΘA = {θn: n ∈ A} of Θhas been investigated by Chousionis, Leykekhman and Urbański in [7, 8] for different subsets A ⊆ N. In particular, they considered the set of powers Pq = {qn: n ∈ N}, q ≥ 2 and asked among other questions if it has a full dimension. We give an affirmative answer to their question in Chapter 4. A significant part of this thesis is devoted to answering open questions and extending results from [7].

It is well known that the Hausdorff dimension of the invariant set of a subsystem of Θ can be analysed using Perron-Frobenius operators, see [7, 8, 16, 17, 20, 21, 23–25, 28, 39]. Interestingly, some of the properties of the dimension spectrum such as compactness and perfectness can be studied and analysed at a purely operator-theoretical level, by considering the spectrum of a family of Perron-Frobenius type operators, without having to make a direct reference to the dimension spectrum of the iterated function. One objective of this thesis is to establish this observation.

The structure of the thesis will be as follows: In Chapter 1, we recall the relevant theory of general iterated function systems and summarise known results on the dimension spectrum.

In Chapter 2, we analyse Perron-Frobenius operators and introduce the notion of dimension spectrum of a class of Perron-Frobenius operators. We prove a new result Theorem 2.30 showing that, in general, the dimension spectrum of these classes of Perron-Frobenius operators is a compact and perfect set, and we relate this result to existing results in the literature concerning the dimension spectrum of iterated function systems.

In Chapter 3, we recall the connection between the Hausdorff dimension of invariant sets and Perron-Frobeinus operators, and the computational methods of Falk and Nussbaum [16, 17] to find rigorous estimates for the Hausdorff dimension of continued fraction expansions. We will also present some new ways of estimating the Hausdorff dimensions for certain families of continued fraction expansions, see Theorem 3.6.

Chapter 4 is devoted to the study of concrete infinite subsystems of continued fraction expansions and extending the results from [7] by using the theory developed in Chapter 2 to analyse the dimension spectrum. We prove that the set of powers Pq = {qn: n ∈ N} has a full dimension spectrum for q ≥ 2, see Theorem 4.1, answering a question of Chousionis, Leykekhman and Urbański [7]. In contrast, in Theorem 4.3, we show that the dimension spectrum of P∗ q = {qn: n ∈ N} ∪ {1} has infinitely many gaps and regions where it is nowhere dense. We also investigate the case where the infinite subsystem is generated by monomials, Mq = {nq: n ∈ N}, see Theorem 4.4. For these sets, we show that the dimension spectrum of Mq is, in general, a finite union of disjoint closed intervals. In particular, we show that it is full, that is, consists of a single closed interval, for q ∈ {1,2,3,4,5}. It consists of two intervals for q ∈{6,7,8}, three intervals for q ∈ {9,10,11,12} and four intervals for q = 19, see Theorem 4.4. We also study the case where A = {22n: n ∈ N}, and prove that its dimension spectrum is a nowhere dense set, Theorem 4.6. Most of these results are written up in [6].

Finally, in Chapter 5, we revisit the set P∗ q, and provide some partial results towards obtaining a complete understanding of its dimension spectrum. We also present several open problems that lie beyond the scope of our methods.