This minicourse presents the proof of the following statement: Any non-atomic self-conformal measure with respect to a non-affine real-analytic iterated function system (IFS) on R has polynomial Fourier decay, hence positive Fourier dimension.

The lectures will synthesize the main ideas and results from my joint works with Federico Rodríguez Hertz and Zhiren Wang, and with Yuanyang Chang, Meng Wu, and Yu-Liang Wu.

Talk 1: Introduction and statement of the main result

We begin with background on the Fourier decay problem for stationary measures and outline the main goal of the minicourse. We will then discuss a Rigidity Theorem for real-analytic IFSs: Either the IFS satisfies a version of Dolgopyat's UNI (Uniform Non-Integrability) condition, or it is simultaneously linearizable by a real-analytic map.  

Talk 2: The linearizable case and L^2 flattening

The main goal of this talk is to handle the linearizable case. Specifically, we will prove that a C^2 strictly convex image of a self-similar measure has positive Fourier dimension. We will also introduce our main technical tool: the fact that these measures exhibit $L^2$ flattening, which will be used as a black box.

Talk 3: The non-linearizable case and exponentially fast Local Limit Theorems

This talk addresses the non-linearizable case. We will show that if a real-analytic IFS cannot be simultaneously linearized to a self-similar system, then all of its self-conformal measures have positive Fourier dimension. The proof relies on a crucial mechanism (used here as a black box): certain random walks driven by the derivative cocycle equidistribute exponentially fast towards a smooth measure.

Talk 4: Exponentially fast LLT, spectral gaps, and a cocycle version of Dolgopyat's method

In the final talk, we will unwrap the black box used in the previous lecture. We will explain how the exponentially fast Local Limit Theorem follows from a spectral gap for the twisted transfer operator corresponding to the derivative cocycle. Finally, we will outline the proof of this spectral gap using a cocycle version of Dolgopyat’s method.

Classical continued fractions provide the best rational approximations of real numbers. Multidimensional continued fractions are expected to produce simultaneously rational approximations with the same denominator. The main advantage of most classical unimodular continued fractions is that they can be described as dynamical systems whose ergodic study is well understood.  In this lecture, we discuss convergence properties in terms of convergence  of infinite products of matrices with nonnegative entries.

The quality of approximation in higher dimensions is  then governed by Lyapounov exponents which describe the asymptotic behaviour of the singular values of large products of random matrices, under the ergodic hypothesis.

In collaboration with W. Steiner and J. Thuswaldner, we have noticed experimentally that the second Lyapounov exponent seems to be positive in higher dimension for the most classical continued fraction algorithms, which prevents their strong convergence. We discuss possible ways to confirm these experiments.

We also discus applications to symbolic dynamics. 

TBA

 1. Basic continued fractions, ordinary irrationality measure function. Best approximations and Lagrange theorem. Different types of continued fractions algorithms.  Geometric aspects of continued fractions' algorithms. Minkowski diagonal continued fraction.

2.  Lagrange, Dirichlet and Minkowski spectra. Second order irrationality measure functions and related spectra.

 3. Problems of distribution of rationals: Franel theorem and Minkowski  question mark function.  Singular distribution functions. Fixed point problem.

 4. Multidimensional generalisations, theory of patterns, Diophantine exponents.  Geometry multidimensional  best approximations. Basic concepts  of parametric geometry of numbers. 

5. Oscillation of irrationality measure functions. Kahn-Moshchevitin phenomenon.  Permutations of values of several irrationality measure functions. Extremal examples.

 6. Approximation by algebraic numbers, Rational approximation to a real number and its square. Roy's theorem and  self-similar constructions. Extremal numbers.  Approximations to a number, its square and its cube.

In this mini-course we explore the structure of a one-parameter family of continued fractions known as alpha-continued fractions, introduced by Nakada and coauthors in the 1980s.  The dynamics of the one-dimensional maps associated to them exhibit a very rich bifurcation theory.  Using the dynamical notion of “matching”, we will be able to describe very explicitly the structure of their parameter space, and extract information about their entropy.  Moreover, it turns out that the combinatorics of this parameter space is isomorphic in a very precise sense to the combinatorics of the family of real quadratic polynomials. This sets up a dictionary between alpha-continued fractions and one-dimensional complex dynamics, with connections to the structure of the real slice of the Mandelbrot set. 

The work presented in this mini-course is mostly joint with Carlo Carminati.