Alpha-continued fractions, matchings and complex dynamics (Lecture 1)

Simons Semester Continued Fractions, Fractals, Ergodic theory and Dynamics Będlewo Conference Center, Poland 26 April - 1 May, 2026 Lecture by: Giulio Tiozzo “Alpha-continued fractions, matchings and complex dynamics” 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 This lecture was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).