Higher-order asymptotic formulae for periodic orbits of 3-dimensional volume-preserving Anosov flows and the Foulon conjecture
Abstract
Let Xt be a volume-preserving Anosov flow of class C7 on a closed $3$-manifold. We consider a sequence of periodic orbits which shadow a homoclinic loop of a fixed periodic orbit γ, and derive higher-order asymptotic expansions for their periods Tn and for the logarithms of their unstable multipliers logλn. We give geometric interpretations of the coefficients in these expansions, in particular in terms of the Anosov cocycle (à la Hurder-Katok) and the longitudinal Anosov cocycle (à la Foulon-Hasselblatt), evaluated on γ. Following the idea of De Simoi-Leguil-Vinhage-Yang, we compare these asymptotics when volume is the measure of maximal entropy. In this case, the equilibrium-state characterizations of volume and the measure of maximal entropy synchronize periods and exponents: logλn=hTn. Matching the coefficients of the two asymptotic expansions forces the cohomology classes of both cocycles to vanish. Then the work of Foulon-Hasselblatt on the longitudinal Anosov cocycle, together with Foulon's work on the contact case, yields the Foulon Conjecture for flows of class C7: if volume is the measure of maximal entropy, then, up to a finite cover, Xt is smoothly conjugate either to a constant-roof suspension of a hyperbolic automorphism of the $2$-torus or to a finite cover of the constant-speed geodesic flow on a closed hyperbolic surface.
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BibTeX
@misc{gogolev2026higherorder,
title = {{Higher-order asymptotic formulae for periodic orbits of 3-dimensional volume-preserving Anosov flows and the Foulon conjecture}},
author = {Andrey Gogolev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31823},
eprint = {2610.31823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31823},
}
RIS
TY - UNPB
TI - Higher-order asymptotic formulae for periodic orbits of 3-dimensional volume-preserving Anosov flows and the Foulon conjecture
AU - Andrey Gogolev
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31823
UR - https://arxiv.org/abs/2610.31823
AB - Let X^t be a volume-preserving Anosov flow of class C^7 on a closed $3$-manifold. We consider a sequence of periodic orbits which shadow a homoclinic loop of a fixed periodic orbit γ, and derive higher-order asymptotic expansions for their periods T_n and for the logarithms of their unstable multipliers logλ_n. We give geometric interpretations of the coefficients in these expansions, in particular in terms of the Anosov cocycle (à la Hurder-Katok) and the longitudinal Anosov cocycle (à la Foulon-Hasselblatt), evaluated on γ. Following the idea of De Simoi-Leguil-Vinhage-Yang, we compare these asymptotics when volume is the measure of maximal entropy. In this case, the equilibrium-state characterizations of volume and the measure of maximal entropy synchronize periods and exponents: logλ_n=hT_n. Matching the coefficients of the two asymptotic expansions forces the cohomology classes of both cocycles to vanish. Then the work of Foulon-Hasselblatt on the longitudinal Anosov cocycle, together with Foulon's work on the contact case, yields the Foulon Conjecture for flows of class C^7: if volume is the measure of maximal entropy, then, up to a finite cover, X^t is smoothly conjugate either to a constant-roof suspension of a hyperbolic automorphism of the $2$-torus or to a finite cover of the constant-speed geodesic flow on a closed hyperbolic surface.
ER -