Variational Principles for r-Neutralized Pfister-Sullivan Entropy
Abstract
In this paper, we study r-neutralized Pfister--Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On r-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower r-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper r-neutralized topological entropy (iii) We obtain upper bounds for r-neutralized PS entropy under different smoothness assumptions and lower bounds for C1+α diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For C∞ diffeomorphisms and C1 Anosov diffeomorphisms, the r-neutralized PS entropies equal hμ(f)+rdim M for every ergodic measure μ. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for C2 surface diffeomorphisms.
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BibTeX
@misc{chen2026variational,
title = {{Variational Principles for $r$-Neutralized Pfister-Sullivan Entropy}},
author = {Ercai Chen and Yunxiang Xie and Xiaoyao Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11639},
eprint = {2610.11639},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11639},
}
RIS
TY - UNPB
TI - Variational Principles for r-Neutralized Pfister-Sullivan Entropy
AU - Ercai Chen
AU - Yunxiang Xie
AU - Xiaoyao Zhou
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11639
UR - https://arxiv.org/abs/2610.11639
AB - In this paper, we study r-neutralized Pfister--Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On r-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower r-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper r-neutralized topological entropy (iii) We obtain upper bounds for r-neutralized PS entropy under different smoothness assumptions and lower bounds for C^1+α diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For C^∞ diffeomorphisms and C^1 Anosov diffeomorphisms, the r-neutralized PS entropies equal h_μ(f)+rdim M for every ergodic measure μ. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for C^2 surface diffeomorphisms.
ER -