×Time: 14.15 on 2 May, 2019
Vesselin Dimitrov (University of Cambridge)
Exponential arithmetic growth in meromorphic dynamics
A conjecture of Silverman (as extended to any global field $K$) states that every Zariski-dense forward orbit $(f^n(P))_{n \in \mathbb{N}}$ under a dominant rational self-map $f : X \dashrightarrow X$ of an $N$-dimensional projective variety $X$ over $K$ has its height $h(f^n(P))$ growing at the maximal possible exponential rate $\lambda_1(f)$, the first dynamic degree of $f$. I will describe some recent progress around this problem, limiting for the main part to the case of polynomial mappings of affine space, and focusing mostly on cases where an exponential growth can be established for the height along all Zariski-dense orbits. The latter includes all additive polynomial mappings of affine space over $\mathbb{F}_q(t)$ (building on Yu's zero estimate from transcendence theory in positive characteristic), as well as all maps of large topological degree (building on work of Habegger from his paper Special points in fibered powers of elliptic surfaces, and on a theorem of Bell, Ghioca and Tucker on the dynamical Mordell-Lang problem). The proof in the former case yields also a close counterpart, for abelian $t$-modules in function field arithmetic, of Masser's "polynomial in 1/D" canonical height lower bound on abelian varieties. Lastly, and time permitting, I will supplement the exponential growth discussion with a lower bound on height growth for an arbitrary polynomial mapping. This bound comes from a partial progress on a conjecture of Ruzsa, and it is weaker than exponential, yet strong enough to yield as application an "unbounded or eventually periodic" dichotomy for the degree sequence of the iterates of a polynomial mapping of $\mathbb{A}_k^N$ over any field. The dichotomy extends a result of Favre and Jonsson from $N = 2$ to arbitrary dimension.