Space spanned by characteristic exponents

Abstract

We prove several rigidity results on multiplier spectrum 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 $d\ge 2$, the $\mathbb{Q}$-vector space generated by all the (finite) characteristic exponents 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 spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on $({\mathbb{P}}^1{)}^N,N\ge 1$.

revised version, 35 pages