A positive proportion Livshits theorem

Dilsavor, Caleb; Marshall Reber, James

doi:10.1090/proc/16880

A positive proportion Livshits theorem
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by Caleb Dilsavor and James Marshall Reber;

Proc. Amer. Math. Soc. 152 (2024), 4729-4744

DOI: https://doi.org/10.1090/proc/16880

Published electronically: September 24, 2024

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Abstract:

Given a transitive Anosov diffeomorphism or flow on a closed connected Riemannian manifold $M$, the Livshits theorem states that a Hölder function $\varphi : M \to \mathbb {R}$ is a coboundary if all of its periods vanish. We explain how a finer statistical understanding of the distribution of these periods can be used to obtain a stronger version of the classical Livshits theorem where one only has to check that the periods of $\varphi$ vanish on a set of positive asymptotic upper density. We also include a strengthening of the nonpositive Livshits theorem.

References

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 224110
Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 399413, DOI 10.1007/BF01762666
K. Burns and A. Katok, Manifolds with nonpositive curvature, Ergodic Theory Dynam. Systems 5 (1985), no. 2, 307–317. MR 796758, DOI 10.1017/S0143385700002935
S. Cantrell and E. Reyes, Marked length spectrum rigidity from rigidity on subsets, arXiv:2304.13209, 2023.
Stephen Cantrell and Richard Sharp, A central limit theorem for periodic orbits of hyperbolic flows, Dyn. Syst. 36 (2021), no. 1, 142–153. MR 4241203, DOI 10.1080/14689367.2020.1849030
Zaqueu Coelho and William Parry, Central limit asymptotics for shifts of finite type, Israel J. Math. 69 (1990), no. 2, 235–249. MR 1045376, DOI 10.1007/BF02937307
Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150–169. MR 1036134, DOI 10.1007/BF02566599
Christopher B. Croke, Nurlan S. Dairbekov, and Vladimir A. Sharafutdinov, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352 (2000), no. 9, 3937–3956. MR 1694283, DOI 10.1090/S0002-9947-00-02532-0
Françoise Dal’bo, Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), no. 2, 199–221 (French, with English and French summaries). MR 1703039, DOI 10.1007/BF01235869
Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097–1114. MR 1653299, DOI 10.1017/S0143385798117431
A. Gogolev and F. Rodriguez Hertz, Smooth rigidity for $3$-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity, arXiv:2210.02295, 2022.
Colin Guillarmou, Gerhard Knieper, and Thibault Lefeuvre, Geodesic stretch, pressure metric and marked length spectrum rigidity, Ergodic Theory Dynam. Systems 42 (2022), no. 3, 974–1022. MR 4374964, DOI 10.1017/etds.2021.75
C. Guillarmou, T. Lefeuvre, and G. Paternain, Marked length spectrum rigidity for Anosov surfaces, arXiv:2303.12007, 2023.
A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
A. Livshits, Cohomology of dynamical systems, Math. USSR-Ivestiya, 6 (1972), no. 6, 1278.
Artur O. Lopes and Philippe Thieullen, Sub-actions for Anosov diffeomorphisms, Astérisque 287 (2003), xix, 135–146 (English, with English and French summaries). Geometric methods in dynamics. II. MR 2040005
A. O. Lopes and Ph. Thieullen, Sub-actions for Anosov flows, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 605–628. MR 2129112, DOI 10.1017/S0143385704000732
Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, 151–162 (French). MR 1038361, DOI 10.2307/1971511
William Parry, Equilibrium states and weighted uniform distribution of closed orbits, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 617–625. MR 970574, DOI 10.1007/BFb0082850
William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
Vesselin Petkov and Luchezar Stoyanov, Ruelle transfer operators with two complex parameters and applications, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6413–6451. MR 3543593, DOI 10.3934/dcds.2016077
Joseph F. Plante, Anosov flows, Amer. J. Math. 94 (1972), 729–754. MR 377930, DOI 10.2307/2373755
Mark Pollicott and Richard Sharp, Error terms for closed orbits of hyperbolic flows, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 545–562. MR 1827118, DOI 10.1017/S0143385701001274
M. Ratner, The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature, Israel J. Math. 16 (1973), 181–197. MR 333121, DOI 10.1007/BF02757869
Richard Schwartz and Richard Sharp, The correlation of length spectra of two hyperbolic surfaces, Comm. Math. Phys. 153 (1993), no. 2, 423–430. MR 1218308, DOI 10.1007/BF02096650
N. Sawyer, Partial marked length spectrum rigidity of negatively curved surfaces, PhD thesis, Wesleyan University, 2020.
Simon Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 4, 445–484. MR 1404318, DOI 10.1016/S0294-1449(16)30110-X