This school is a part of the Simons Semester Continued Fractions, Fractals, Ergodic theory and Dynamics. The conference will be a central event of the Semester, hosting both leaders of the field and young researchers, allowing for intense interactions. It will bring together the main topics of focus of the Semester.

Diophantine approximation is one of the central fields of number theory, dealing with the speed of approximation of irrational numbers with rational numbers. It is intimately linked with ergodic theory and fractal geometry, with continued fractions being a major tool in each of those fields. For instance, it turned out that studying continued fraction expansion with bounded partial quotients can help one to gain understanding of the structure of Lagrange and Markov spectra, that give “best constants” in approximation of irrational numbers. More generally, the sets defined in terms of continued fractions of their elements have very intricate, fractal structure, and their Hausdorff dimension plays an important role in Number-theoretic applications - in particular in Zaremba conjecture and in the problem of counting spanning trees in planar graphs. In holomorphic dynamics, a classical result by Brjuno-Yoccoz states that for a quadratic polynomial with a rationally indifferent fixed point the dynamics near this point is analytically conjugate to a rotation if and only if the associated Brjuno function evaluated at the rotation angle is finite. Brjuno functions, used as a tool in holomorphic dynamics, on their own present an independent interest. In addition they have deep connections with Kolmogorov–Arnold–Moser (KAM) theory of quasiperiodic motions for Hamiltonian systems, through which continued fractions appear in astronomy when studying the stability of motion of objects in space such as planets and asteroids.

Confirmed invited speakers:

• Anna Zdunik

• Antti Käenmäki

• Ariel Rapaport

• Bruno Martin

• Charlene Kalle

• Claire Burrin

• Claudio Bonanno

• Demi Allen

• Dmitry Kleinbock

• Faustin Adiceam

• Felipe A . Ramirez

• Han Yu

• Hitoshi Nakada

• Jonathan Fraser

• Kate Stange

• Katrin Gelfert

• Lingmin Liao

• Mariusz Urbański

• Michel Balazard

• Natalia Jurga

• Nikolay Moshchevitin 

• Polina Vytnova

• Sandro Vaienti

• Sanju Velani

• Sascha Troscheit

• Seonhee Lim

• Simon Baker

• Stefano Marmi

• Stéphane Jaffard

• Vadim Kaloshin

• Valérie Berthé

• Yann Bugeaud