arXiv:2610.31823v1 Announce Type: new Abstract: 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 $\gamma$, and derive higher-order asymptotic expansions for their periods $T_n$ and for the logarithms of their unstable multipliers $\log\lambda_n$. We give geometric interpretations of the coefficients in these expansions, in particular in terms of the Anosov cocycle (\`a la Hurder-Katok) and the longitudinal Anosov cocycle (\`a la Foulon-Hasselblatt), evaluated on $\gamma$. 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: $\log\lambda_n=hT_n$. 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.
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