Kevin Pilgrim
University of Indiana
https://pilgrim.pages.iu.edu/
Date(s) : 22/09/2023 iCal
11 h 00 min - 12 h 00 min
Real bimodal quadratic rational maps: moduli space and entropy (with K. Filom and S. Kang)
Abstract: Bruin-van Strien and Kozlovski showed that for multimodal self-maps $f$ of the unit interval, the function $f \mapsto h(f)$ sending $f$ to its topological entropy is monotone. K. Filom and I showed that for interval maps arising from real bimodal quadratic rational maps, this monotonicity fails. A key ingredient in our proof is an analysis of a family $f_{p/q}, p/q \in \mathbb{Q}/\mathbb{Z}$ of critically finite maps on which the dynamics on the postcritical set is conjugate to the rotation $x \mapsto x+p \bmod q$ on $\mathbb{Z}/q\mathbb{Z}$, where $x=0$ and $x=1$ correspond to the two critical points. The recent PhD thesis of S. Kang constructs a piecewise-linear (PL) copy of the well-known Farey tree whose vertices are expanding PL quotients of the $f_{p/q}$’s. This PL model, conjecturally, sheds light on the moduli space of the real quadratic bimodal family, and on the variation of entropy among such maps.
Emplacement
FRUMAM, St Charles (2ème étage)
Catégories