Macroscopic return-time spectra and Hausdorff dimension transitions

Abstract

Can the complete macroscopic spectrum of return times determine the Hausdorff dimension of the corresponding set of initial points? We classify all common lower and upper macroscopic intermediate-dimension spectra of zero-density integer sets and realize every admissible spectrum on a full-dimensional family of zero-frequency orbits under logarithmic-power bounds on inverse-branch contraction. For first-return branches with stretched-exponential contraction $e^{-n^δ}$, $0<δ<1$, complete spectrum levels exhibit a sharp zero/full dimension transition at the counting endpoint $δ$. At the critical exponent, exact laws $N_y(t)\sim c_0t^δ(\log t)^β$ yield zero dimension for $β\le0$ and full dimension for $β>0$, even with the complete spectrum fixed. An explicit two-branch intermittent interval map realizes this transition through genuine first returns. Above the critical exponent, imposing an exact counting law and a complete spectrum does not reduce the dimension of an intrinsic mean-energy level in an affine return tower. For nonlinear branches, we obtain quantitative bounds and an exact conditional-pressure formula under thermodynamic regularity assumptions, verified for an explicit non-affine family.