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arXiv · 2308.00289

Space spanned by characteristic exponents

Zhuchao Ji·Junyi Xie·Geng-Rui Zhang
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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 $(\mathbb{P}^1)^N, N\geq 1$.

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Zhuchao Ji, Junyi Xie, Geng-Rui Zhang. 2023-08-01. Space spanned by characteristic exponents. https://doi.org/10.1007/s00208-026-03361-4

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