α-BS dimension on subsets

2025/12/15 by Zhumin Ding, Rui Yang, Ding, Zhumin +3
Mathematics · Materials Science · Computer Science · #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.2512.13075

Abstract

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

Citations

• On r-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum
• Variational principle for neutralized Bowen topological entropy
• Neutralized local entropy, and dimension bounds for invariant measurs
• Exponential state estimation, entropy and Lyapunov exponents
• Upper estimates for stable dimensions of fractal sets with variable number of foldings
• Variational Principles For BS Dimension of Subsets
• Variational principles for topological entropies of subsets
• Variational principles for topological entropies of subsets
• On the topological entropy of saturated sets
• Topological entropy for divergence points
• On the variational principle for the topological entropy of certain non-compact sets
• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Repellers for real analytic maps
• Topological entropy for noncompact sets

Related

• Variational principles for topological pressures on subsets
• Topological pressure dimension for almost additive potentials
• Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure
• A variational principle for topological pressure for certain non-compact sets
• Metric mean dimension, Hölder regularity and Assouad spectrum