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arXiv · 2508.17800

Variations of topological theory and ergodic theory via gap function in non-uniform specification

Wanshan Lin·Xueting Tian·Chenwei Yu
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Abstract

In our previous work [43], we studied qualitative differences between specification and nonuniform specification. In this paper, we investigate quantitative variations depending on the gap function. We first obtain lower bounds for the Bowen topological entropy of irregular sets in terms of the lower linear growth of gap function. In contrast, we prove that every non-empty over-saturated set has full packing topological entropy under non-uniform specification. We also establish a quantitative lower bound for the Bowen topological entropy of transitive points under non-uniform specification, and a lower bound for the exponential growth of periodic orbits under its periodic version. Besides, we construct symbolic systems with non-uniform specification with a given gap growth rate which contain an arbitrary subshift. These systems show that certain bounds concerning irregular sets and periodic orbits are optimal. They also show that positive linear gap growth may destroy full Bowen entropy of transitive points, the conditional variational principle, the intermediate entropy and pressure properties, and the genericity of continuous functions whose unique maximizing measure has zero entropy.

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Wanshan Lin, Xueting Tian, Chenwei Yu. 2026-09-20. Variations of topological theory and ergodic theory via gap function in non-uniform specification. https://arxiv.org/abs/2508.17800

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