$\hat{I}\pm$-BS dimension on subsets

Abstract

We aim to investigate the dimension theory of $\hat{I}\pm$-pressure-like quantities. By means of the Carath$\acute{\mathrm{e}}$odory-Pesin structure, we define $\hat{I}\pm$-BS dimension and $\hat{I}\pm$-Pesin topological pressure on subsets using $\hat{I}\pm$-Bowen metric $$d_n^{\hat{I}}\pm (x,y)=\underset{0\le i\le n-1}{\max }{e}^{\hat{I}\pm i}d(f^{i}x,f^{i}y),$$ where $\hat{I}\pm \ge 0$. Specifically, we show that $\hat{I}\pm$-BS dimension and $\hat{I}\pm$-Pesin topological pressure are related by a Bowen's equation. Inspired by the classical Brin-Katok entropy, we introduce the notion of $\hat{I}\pm$-local Brin-Katok entropy, and establish a variational principle for $\hat{I}\pm$-BS dimension on compact subsets in terms of $\hat{I}\pm$-local Brin-Katok entropy. Besides, for subshifts of finite type, we prove that $\hat{I}\pm$-Bowen topological entropy is closely related to spectral radius and Hausdorff dimension.

26 pages