1. Space spanned by characteristic exponents
- Author
-
Ji, Zhuchao, Xie, Junyi, and Zhang, Geng-Rui
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,Primary 37P35, Secondary 37F10, 11G35 - Abstract
We prove several rigidity results on multiplier and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by the characteristic exponents (that are not $-\infty$) of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectrum. Finally as an application of our result, we get a new proof of the Zariski dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$., Comment: 27 pages
- Published
- 2023