Uniform approximation problems of expanding Markov maps

作者：He, Yubin^*; Liao, Lingmin

来源：Ergodic Theory and Dynamical Systems, 2024, 44(1): 159-183.

DOI：10.1017/etds.2023.10

• uniform Diophantine approximation
• expanding Markov map
• Hausdorff dimension
• Gibbs measure
• multifractal spectrum

Let T : [0, 1] -> [0, 1] be an expanding Markov map with a finite partition. Let mu(phi) be the invariant Gibbs measure associated with a Holder continuous potential phi. For x is an element of[0, 1] and kappa > 0, we investigate the size of the uniform approximation set @@@ U-kappa (x) := {y is an element of[0, 1] : for all N >> 1, there exists n <= N, such that |T-n x - y| < N-kappa}. @@@ The critical value of kappa such that dimH U-kappa (x) = 1 for mu(phi)-almost every (a.e.) x is proven to be 1/alpha(max), where alpha(max) = - integral phi d mu(max)/ integral log |T '| d mu(max) and mu(max) is the Gibbs measure associated with the potential - log |T '|. Moreover, when kappa > 1/alpha(max), we show that for mu(phi)-a.e. x, the Hausdorff dimension of U-kappa (x) agrees with the multifractal spectrum of mu(phi).

• 单位
  武汉大学