Macroscopic return-time spectra and Hausdorff dimension transitions

@misc{li2026macroscopic,
  title = {{Macroscopic return-time spectra and Hausdorff dimension transitions}},
  author = {Jinjun Li},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06416},
  eprint = {2611.06416},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06416},
}

TY  - UNPB
TI  - Macroscopic return-time spectra and Hausdorff dimension transitions
AU  - Jinjun Li
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06416
UR  - https://arxiv.org/abs/2611.06416
AB  - 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)∼ c_0t^δ(log t)^β yield zero dimension for β≤0 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.
ER  -