On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

• DOI: 10.48550/arxiv.2405.03213
• Publication Date: 2024-05-06
• Authors: Zhou Feng
• Keywords: FOS: Mathematics; Dynamical Systems (math.DS); Mathematics - Dynamical Systems; 28A80, 37D35

Let $ K be a compact subset of the $d$-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose symbolic coding $K$ satisfies weak specification. When \leq 2 $, we prove that following three statements are equivalent: (A) Hausdorff and box dimensions coincide; (B) respect to some gauge function, measure is positive finite; (C) dimension maximal entropy on attains $. \geq 3 find examples in which does not hold but holds, new phenomenon appearing planar cases. Through different probabilistic approach, establish equivalence for Bedford-McMullen sponges.