$\alpha$-BS dimension on subsets

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