The Ultimate Ishii-Pastur Theorem for Whole-Line Ergodic Block Jacobi Operators

Zhenfu Wang; Bei Zhang; Qi Zhou

The Ultimate Ishii-Pastur Theorem for Whole-Line Ergodic Block Jacobi Operators

Spectral Theory 2026-07-28 v1 Mathematical Physics Dynamical Systems

Authors: Zhenfu Wang , Bei Zhang , Qi Zhou

Abstract

We establish an ultimate Ishii-Pastur theorem for whole-line ergodic block Jacobi operators. We prove that, for almost every realization, the restriction of a maximal spectral measure to the region where the smallest nonnegative Lyapunov exponent is strictly positive is carried by a Borel set of zero capacity. In the scalar case, this settles the whole-line problem formulated by Damanik and Fillman \cite[Problem~4.7.18]{Damanik-Fillman-book}.

Keywords

Ishii-Pastur theorem ergodic block Jacobi operators Lyapunov exponents spectral measures capacity

Cite

@article{arxiv.2607.25808,
  title  = {The Ultimate Ishii-Pastur Theorem for Whole-Line Ergodic Block Jacobi Operators},
  author = {Zhenfu Wang and Bei Zhang and Qi Zhou},
  journal= {arXiv preprint arXiv:2607.25808},
  year   = {2026}
}