Friday December 1st, 2023 | |
| Time: | 2:30 PM - 3:30 AM |
| Title: | Real bimodal quadratic rational maps: moduli space and entropy (with K. Filom and S. Kang) |
| Speaker: | Kevin Pilgrim, Indiana University Bloomington |
| Location: | Math P-131 |
| Abstract: | |
| Bruin-van Strien and Kozlovski showed that for multimodal self-maps $f$ of the unit interval, the function $f → h(f)$ 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 $f_{p/q}, p/q ∈ \mathbb{Q}/\mathbb{Z}$ of critically finite maps on which the dynamics on the postcritical set is conjugate to the rotation $x → x+p b \mod q$ on $\mathbb{R}/\mathbb{Z}$, 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. | |