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arXiv · 2606.11528

A dynamical proof of non-arithmeticity of Jordan spectra

Hee Oh·Pratyush Sarkar
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Abstract

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

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Hee Oh, Pratyush Sarkar. 2026-06-10. A dynamical proof of non-arithmeticity of Jordan spectra. https://arxiv.org/abs/2606.11528

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