Real bimodal quadratic rational maps: moduli space and entropy (with K. Filom and S. Kang)
Date(s) : 22/09/2023 iCal
11h00 - 12h00
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 $mathbb{Z}/qmathbb{Z}$, 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
Saint-Charles - FRUMAM (2ème étage)
Catégories
