Empirical variational principles for preimage entropies
Abstract
Preimage entropy measures the complexity generated by the inverse-image structure of a non-invertible dynamical system. For a continuous map f:X→ X on a compact metric space, Hurley's pointwise topological preimage entropies hm(f) and hp(f) are natural invariants measuring the complexity of preimage sets. The question of whether they admit unconditional variational principles in terms of suitable measure-theoretic counterparts remains open. In this paper we resolve it by using empirical metric preimage entropies h*m,μ(f) and h*p,μ(f), defined by restricting preimage fibers to orbit segments whose empirical measures are close to a prescribed invariant measure μ. We prove the variational principles hm(f)=supμ∈ℳf(X)h*m,μ(f), hp(f)=supμ∈ℳf(X)h*p,μ(f) for every continuous map on a compact metric space. We also show that, in general, the set of all invariant measures in these formulas cannot be replaced by the set of ergodic invariant measures. We then compare the empirical entropies with the pointwise metric preimage entropy hm,μ(f). For every ergodic invariant measure μ, we prove h*p,μ(f)≥ hm,μ(f). Moreover, if f has uniform separation of preimages, then for every ergodic invariant measure μ, h*m,μ(f)=h*p,μ(f)=hm,μ(f). Examples show that hm,μ(f) is not comparable with the empirical quantities in general. We further introduce a resolving-partition property, weaker than uniform separation of preimages, under which the variational principle for hm(f) and hm,μ(f) holds. Finally, we establish corresponding variational principles for preimage pressure and give an example showing that uniform separation of preimages does not imply forward expansiveness.
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BibTeX
@misc{wang2026empirical,
title = {{Empirical variational principles for preimage entropies}},
author = {Tao Wang and Yi Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00655},
eprint = {2609.00655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00655},
}
RIS
TY - UNPB
TI - Empirical variational principles for preimage entropies
AU - Tao Wang
AU - Yi Yang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00655
UR - https://arxiv.org/abs/2609.00655
AB - Preimage entropy measures the complexity generated by the inverse-image structure of a non-invertible dynamical system. For a continuous map f:X→ X on a compact metric space, Hurley's pointwise topological preimage entropies h_m(f) and h_p(f) are natural invariants measuring the complexity of preimage sets. The question of whether they admit unconditional variational principles in terms of suitable measure-theoretic counterparts remains open. In this paper we resolve it by using empirical metric preimage entropies h^*_m,μ(f) and h^*_p,μ(f), defined by restricting preimage fibers to orbit segments whose empirical measures are close to a prescribed invariant measure μ. We prove the variational principles h_m(f)=sup_μ∈ℳ_f(X)h^*_m,μ(f), h_p(f)=sup_μ∈ℳ_f(X)h^*_p,μ(f) for every continuous map on a compact metric space. We also show that, in general, the set of all invariant measures in these formulas cannot be replaced by the set of ergodic invariant measures. We then compare the empirical entropies with the pointwise metric preimage entropy h_m,μ(f). For every ergodic invariant measure μ, we prove h^*_p,μ(f)≥ h_m,μ(f). Moreover, if f has uniform separation of preimages, then for every ergodic invariant measure μ, h^*_m,μ(f)=h^*_p,μ(f)=h_m,μ(f). Examples show that h_m,μ(f) is not comparable with the empirical quantities in general. We further introduce a resolving-partition property, weaker than uniform separation of preimages, under which the variational principle for h_m(f) and h_m,μ(f) holds. Finally, we establish corresponding variational principles for preimage pressure and give an example showing that uniform separation of preimages does not imply forward expansiveness.
ER -