Space spanned by characteristic exponents

TL;DR

This paper establishes new rigidity results linking characteristic exponents and length spectra of rational maps, proving infinite-dimensionality of certain vector spaces and characterizing postcritically finite maps, with applications to the Zariski-dense orbit conjecture.

Contribution

It proves the infinite-dimensionality of the space generated by characteristic exponents for non-exceptional rational maps and characterizes postcritically finite maps via length spectra, extending prior conjectures.

Findings

Infinite-dimensional space of characteristic exponents for rational maps

Characterization of postcritically finite maps through length spectra

New proof of the Zariski-dense orbit conjecture

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 $d\geq2$, 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 $(\mathbb{P}^1)^N, N\geq 1$.