The “transversality method” has been developed to study parametrised families of fractal sets and measures. In this minilecture, we will give a brief overview of the classical and most recent results. We will then study the method’s applications to orthogonal projections, iterated function systems, and stationary measures to investigate dimension, absolute continuity, and other regularity properties, and we will verify the condition in several cases. 

The closed simple geodesics in the modular torus correspond to the quadratic Markov numbers, whose continued fractions are periodic Sturmian sequences on the alphabet {1,2}: they are the most badly approximable by rationals, yielding the initial discrete sequence of the Markov spectrum from 1/\sqrt{5} to 1/3.

The non-closed simple geodesics correspond to numbers whose continued fractions are aperiodic Sturmian sequences on the alphabet {1,2}, yielding all numbers with Markov approximation constant 1/3: an application of Schmidt's subspace theorem implies that all these numbers are transcendental.

The aim of the minicourse will be to present those ideas and generalise this picture to simple geodesics in arithmetic surfaces: the strategy will take us on a journey from the mapping class group action on the space of measured laminations, through the symbolic dynamics and complexity of interval exchange transformations, finishing with the diophantine approximation of numbers by the trace fields of arithmetic Fuchsian groups.

This mini-course will be a survey of fundamental problems and classical and modern results of neutral complex dynamics. That is, the dynamics of a holomorphic map of one complex variable with a neutral fixed point; the simplest family of examples is quadratic polynomials $f_t(z)= e^{2\pi i t} z + z^2$. Within this simple formula lies a rich theory. Historically, the development relies heavily on the Diophantine properties of the rotation number $t$. The goal of this mini-course is two-fold: a survey of classical results and an overview of advances in the last 5 years. I will discuss the following topics.

• Linearization problem: A classical result by Brjuno and Yoccoz states that such a map is analytically linearizable near the neutral fixed point if the rotation number $t$ is a Brjuno irrational. I will explain Yoccoz's proof in the framework of sector renormalization.

• Hedgehogs and circle maps: There is a strong connection between attractors of maps with neutral fixed points with the study of circle diffeomorphisms and critical circle maps. I will discuss this connection, following the works of Perez-Marco, Herman, Douady, Ghys, etc.

• Golden mean case: This is one of the most well understood cases. There is a complete understanding of the topology of the Julia set of $f_t$ (Petersen). Even more surprising, there is also a remarkable rigidity, self-similarity, and universality phenomenon (McMullen).

• All irrationals: I will talk about the most recent development that covers all irrationals $t$. The most recent breakthrough was by Dima Dudko and Misha Lyubich on uniform bounds for renormalization. I will talk about various consequences of these bounds on the properties of the full attractor (Mother Hedgehog) of $f_t$ and a generalization of dynamical universality. This last part is based on a joint project by me, Dima, and Misha.

Classical examples of fractal subsets of the real line are “missing digit” sets. Missing digit sets in a fixed base are very well-behaved. But what about sets of numbers with continued fraction expansion that avoid certain digits? We will see that these sets are much more poorly behaved than their fixed base counterparts: for example, their box dimension need not exist. A key tool to understand this phenomenon is a certain 1-Lipschitz function called the branching function which one can associate with general subsets of the real line. This function is ubiquitous in (continuous) incidence geometry but its appearance in the context of dynamically invariant sets is quite surprising.

In 1855, Seidel introduced an arithmetic procedure, called contraction, which—under mild assumptions—allows one to produce from a given generalised continued fraction (GCF) a new GCF whose convergents are any prescribed subsequence of the original GCF-convergents.  Nearly ninety years later, in 1943, Kakutani introduced induced transformations, which accelerate a given dynamical system by only observing the dynamics within a subregion of the domain.  In 1989, Shunji Ito gave an explicit natural extension of what has been called ‘the mother of all continued fractions’—the Farey tent map—which generates ‘slow’ GCF-expansions whose convergents consist of all regular continued fraction (RCF) convergents and so-called mediant RCF-convergents.  In this mini-course, we introduce each of these tools and show how, together, they yield a broad, unifying theory for various continued fraction expansions (including RCFs, Nakada's $\alpha$-continued fractions, and Kraaikamp's $S$-expansions) and their metrical properties. 

This mini-course is devoted to a set of symmetry-based tricks in a theory of continued fractions that proven themselves to be useful over the years, with the main one being the so-called “Folding lemma”. Applications to be discussed include constructing numbers with given irrationality exponent lying in a Cantor set in both real and complex settings, getting diophantine properties of fixed points of Minkowski question mark function, obtaining non-trivial bounds in Zaremba’s conjecture, and even a continued fraction-based proof of Fermat’s theorem on sums of two squares.