PreprintarXiv licencearXiv
Finite-Regularity Counterexamples to Katok's Periodic-Point Conjecture
Abstract
For any r>1 and any nonnegative sequence ω(n)=o(n), we construct a Cr sphere diffeomorphism f with h=htop(f)>0 such that limn→∞1/nlog#Fix(fn)=h, ∑ngeqslant1e−hn+ω(n)#Fix(fn)<∞. Every ergodic invariant probability measure is hyperbolic, and, apart from finitely many fixed points, all periodic points lie in a single topologically mixing homoclinic class of entropy h. This shows that the logarithmic periodic-point lower bound is optimal in finite regularity. In particular, we answer the finite-regularity case of Katok's periodic-point question in the negative.
Related
- Finite-state counterexamples to Ross's second queueing conjecture
- Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples
- Work in progress: A Lean-verified conditional reduction of periodic Navier-Stokes regularity on T^3 to a single integrability hypothesis
- Counterexamples to the Monotone Completeness Conjecture
- Counterexamples to the ordinary Fried conjecture near hyperbolic metrics
BibTeX
@misc{ding2026finiteregularity,
title = {{Finite-Regularity Counterexamples to Katok's Periodic-Point Conjecture}},
author = {Nan Ding and Yao Tong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01980},
eprint = {2611.01980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01980},
}
RIS
TY - UNPB
TI - Finite-Regularity Counterexamples to Katok's Periodic-Point Conjecture
AU - Nan Ding
AU - Yao Tong
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01980
UR - https://arxiv.org/abs/2611.01980
AB - For any r>1 and any nonnegative sequence ω(n)=o(n), we construct a C^r sphere diffeomorphism f with h=h_top(f)>0 such that lim_n→∞1/nlog#Fix(f^n)=h, ∑_ngeqslant1e^−hn+ω(n)#Fix(f^n)<∞. Every ergodic invariant probability measure is hyperbolic, and, apart from finitely many fixed points, all periodic points lie in a single topologically mixing homoclinic class of entropy h. This shows that the logarithmic periodic-point lower bound is optimal in finite regularity. In particular, we answer the finite-regularity case of Katok's periodic-point question in the negative.
ER -