## Abstract

Given an integer \(k \ge 5\), and a \(C^k\) Anosov flow \(\Phi \) on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if \(\Phi \) is \(C^{k-\varepsilon }\)-conjugate to an algebraic flow, for \(\varepsilon >0\) arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.

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## Notes

By Livsic’s theorem, it is sufficient to have a \(C^0\) function \(u{:}\,M \rightarrow {\mathbb {R}}\) such that (2.1) holds.

See for instance Theorem 4.14 in [Bow75].

They are also equal to the eigenvalues of \(D\mathcal {F}_{x_n}^{n+2}\), see for instance Lemma 1 on p.111 of [HZ12].

Recall that \( N^{n} \circ \mathcal {G}(\eta _n,\xi _n)=(\eta _n,\xi _n)\).

Recall that for \(x=(s,\varphi )\in \mathcal {M}\) and \(x'=(s',\varphi '):=\mathcal {F}(s,\varphi )\), we have \(\mathrm {det}\ D_{x}\mathcal {F}=\frac{\cos \varphi }{\cos \varphi '}\). Thus, for any periodic orbit \(\mathcal {O}=(x_1,x_2,\ldots ,x_p)\) of period \(p \ge 2\), we have \(\mathrm {det}\ D_{x_j}\mathcal {F}^p=1\), for \(j \in \{1,\ldots ,p\}\).

Indeed, thanks to the finite horizon assumption, we can choose \(\varphi _0,n_0\) such that \(s_0<1\).

A similar result also holds in the case of open dispersing billiards.

## References

I. Adeboye, H. Bray and D. Constantine. Entropy rigidity and Hilbert volume.

*Discrete Contin. Dyn. Syst.*(4)39 (2019), 1731–1744.F. Bguin, C. Bonatti and B. Yu. Building Anosov flows on 3-manifolds.

*Geom. Topol.*(3)21 (2017), 1837–1930.V. Baladi and M. Demers. On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps, arXiv preprint, arXiv:1807.02330v3.

P. Bálint, J. De Simoi, V. Kaloshin and M. Leguil. Marked Length Spectrum, homoclinic orbits and the geometry of open dispersing billiards.

*Commun. in Math. Phys.*, (3)374 (2020), 1531–1575.G. Besson, G. Courtois and S. Gallot. Entropies et rigidités des espaces localement symétriques de courbure strictement négative.

*Geom. Funct. Anal*. (5)5 (1995), 731–799.R. Bowen.

*Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms*. Lect. Notes in Mathematics vol.**470**(1975), Springer.N. Chernov. Invariant measures for hyperbolic dynamical systems. In

*Handbook of dynamical systems*, Vol.**1**(2002), 321–407, Elsevier Science.N. Chernov and R. Markarian.

*Chaotic Billiards*, Mathematical Surveys and Monographs,**127**, AMS, Providence, RI, 2006 (316 pp.).Y. Colin de Verdière. Sur les longueurs des trajectoires périodiques d’un billard. In: P. Dazord and N. Desolneux-Moulis (eds.),

*Géométrie Symplectique et de Contact : Autour du Théorème de Poincaré-Birkhoff*, Travaux en Cours, Séminaire Sud-Rhodanien de Géométrie III, Herman (1984), 122–139.R. de la Llave. Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems.

*Commun. Math. Phys*. 150 (1992), 289–320.R. de la Llave and R. Moriyón. Invariants for Smooth Conjugacy of Hyperbolic Dynamical Systems. IV.

*Commun. Math. Phys*. 116 (1988), 185–192.A. Delshams, M.S. Gonchenko and S.V. Gonchenko. On dynamics and bifurcations of area-preserving maps with homoclinic tangencies.

*Nonlinearity*, (9)28 (2015), 3027–3071.J. De Simoi, V. Kaloshin and M. Leguil. Marked Length Spectral determination of analytic chaotic billiards with axial symmetries, arXiv preprint, arXiv:1905.00890v3.

M. Field, I. Melbourne and N. Török. Stability of mixing and rapid mixing for hyperbolic flows.

*Annals of Mathematics*166 (2007), 269–291.P. Foulon. Entropy rigidity of Anosov flows in dimension three.

*Ergodic Theory and Dynamical Systems*(4)21 (2001), 1101–1112.P. Foulon and B. Hasselblatt. Contact Anosov flows on hyperbolic 3-manifolds.

*Geom. Topol.*(2)17 (2013), 1225–1252.J. Franks and B. Williams. Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math. 819, Springer, Berlin (1980), 158–174.

P. Gaspard and S.A. Rice. Scattering from a classically chaotic repellor.

*The Journal of Chemical Physics*90 (1989), 2225E. Ghys. Flots d’Anosov dont les feuilletages stables sont différentiables.

*Annales scientifiques de l’É.N.S. 4e série, tome*(2)20 (1987), 251–270.E. Ghys. Rigidité différentiable des groupes fuchsiens, Publications mathématiques de l’I.H.É.S., tome 78 (1993), 163–185.

S.V. Gonchenko, L.P. Shilnikov and D.V. Turaev. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps, Nonlinearity 20 (2007), 241–275.

M. Handel, W. Thurston. Anosov flows on new three manifolds,

*Invent. Math.*(2)59 (1980), 95–103.B. Hasselblatt and A. Katok.

*Introduction to the Modern Theory of Dynamical Systems*(Encyclopedia of Mathematics, 54), Cambridge University Press, 1995.M. Hirsch and C. Pugh.

*Stable Manifolds and Hyperbolic Sets*, Proc. Symp. Pure Math.**14**, American Mathematical Society, Providence, RI, 1970, 133–164.S. Hurder and A. Katok. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows.

*Publications Mathématiques de l’IHÉS*72 (1990), 5–61.H. Hofer and E. Zehnder.

*Symplectic invariants and Hamiltonian dynamics*, Birkhäuser, 2012.A. Katok. Entropy and closed geodesics.

*Ergodic Theory Dynam. Systems*(3-4)2 (1982), 339–365.A.

*Katok*. Four applications of conformal equivalence to geometry and dynamics.*Ergod. Th. & Dynam. Sys.*8 (1988), 139–152.T. Morita. The symbolic representation of billiards without boundary condition.

*Trans. Amer. Math.Soc.*325 (1991), 819–828.W. Parry. Synchronisation of Canonical Measures for Hyperbolic Attractors.

*Commun. Math. Phys.*106 (1986), 267–275.V.M. Petkov and L.N. Stoyanov.

*Geometry of the generalized geodesic flow and inverse spectral problems*, 2nd ed., John Wiley & Sons, Ltd., Chichester, (2017).J.F. Plante. Anosov Flows.

*Amer. J. Math*. 94 (1972), 729–754.S. Sternberg. The Structure of Local Homeomorphisms, III.

*American Journal of Mathematics*, (3)81 (1959), 578–604.D. Stowe. Linearization in two dimensions.

*J. Differential Equations*63 (1986), 183–226.L. Stoyanov. A sharp asymptotic for the lengths of certain scattering rays in the exterior of two convex domains.

*Asymptotic Analysis*(3,4)35, 235–255.L. Stoyanov. An estimate from above of the number of periodic orbits for semi-dispersed billiards.

*Comm. Math. Phys.*124 (1989), 217–227.L. Stoyanov. Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows.

*Amer. J. Math.*123 (2001), 715–759.J. Yang.

*Entropy rigidity for three dimensional volume-preserving Anosov flows*, arXiv preprint, arXiv:1807.02330v3.L.S. Young. What Are SRB Measures, and Which Dynamical Systems Have Them?.

*Journal of Statistical Physics*, (5/6)108 (2002), 733–754.W. Zhang and W. Zhang. Sharpness for \(C^1\) linearization of planar hyperbolic diffeomorphisms.

*J. Differential Equations*257 (2014), 4470–4502.

## Acknowledgements

This paper has gone through several stages, and revisions of the author list. The last two authors would like to thank Cameron Bishop and David Hughes for their work on earlier iterations of these results, in particular related to our understanding of the Anosov class. They would also like to thank the AMS Mathematical Resarch Communities program *Dynamical Systems: Smooth, Symbolic and Measurable*, where several of the initial ideas and work was made on this project. The program also provided travel funds for further collaboration. The authors would like to thank Viviane Baladi, Aaron Brown, Sylvain Crovisier, Andrey Gogolev, Boris Hasselblatt, Carlos Matheus, Rafael Potrie, Federico Rodriguez-Hertz, Ralf Spatzier and Amie Wilkinson for discussions and encouragement on this project. Finally, we would like to acknowledge an unpublished preprint of Jiagang Yang where we learned several useful ideas for the argument in Section 3.3.

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Jacopo De Simoi is supported by the NSERC Discovery grant, Reference No. 502617-2017. Martin Leguil was supported by the ERC Project 692925 NUHGD of Sylvain Crovisier. Yun Yang is supported by a grant from the National Science Foundation (DMS-2000167).

## Bowen–Margulis Measure

### Bowen–Margulis Measure

In this appendix, we assume that \(\Phi \) is a *topologically mixing* smooth Anosov flow on some compact 3-manifold *M*. Let us recall that by a result of Plante [Pla72], \(\Phi \) is topologically mixing if and only if \(E_\Phi ^s\) and \(E_\Phi ^u\) are not jointly integrable. As the measure of maximal entropy is unique, it is given by the construction introduced by Margulis, which we now recall. It is first done by constructing a family of measures \(\nu ^{cu}\) and \(\nu ^s\) defined on leaves of the unstable foliation \(\mathcal {W}_\Phi ^{u}\) and of the stable foliation \(\mathcal {W}_\Phi ^s\), respectively, such that:

where \(h:=h_{\mathrm {top}(\Phi )}>0\) is the topological entropy of \(\Phi \). Moreover, \(\nu ^{u}\) is invariant by holonomies along the leaves of \(\mathcal {W}^s_\Phi \), while \(\nu ^s\) is invariant by holonomies along the leaves of \(\mathcal {W}^{u}_\Phi \). Notice that (A.1) allows \(\nu ^u\) and \(\nu ^s\) to be easily extended to measures \(\nu ^{cu}\) and \(\nu ^{cs}\) on leaves of the weak unstable foliation \(\mathcal {W}_\Phi ^{cu}\) and of the weak stable foliation \(\mathcal {W}_\Phi ^{cs}\), respectively.

The Bowen-Margulis measure \(\mu \) of \(\Phi \) on *M* is then constructed locally using the local product structure of the manifold. That is, at \(x \in M\), choose open neighbourhoods of \(U(x) \subset \mathcal {W}_\Phi ^{cu}(x)\) and \(V(x) \subset \mathcal {W}_\Phi ^s(x)\), so that there is a well-defined map \(\varphi {:}\,O(x) \rightarrow M\) which gives Hölder coordinates on the local product cube \(O(x):=U(x) \times V(x)\subset M\). Fix an arbitrary \(y \in U(x)\). For any open set \(\Omega \subset O(x)\), we let

where for any \(\Omega '\subset V(x)\), we set \(\nu _y(\Omega '):=\nu ^{s} (\{y\}\times \Omega ')\). By the invariance of \(\nu ^{s}\) under weak unstable holonomies, the previous definition is independent of the choice of \(y \in U(x)\) and defines locally the Bowen-Margulis measure \(\mu =\nu ^{cu} \times \nu ^s\).

### Proposition A.1

If the measure of maximal entropy \(\mu \) of \(\Phi \) is absolutely continuous with respect to Lebesgue measure, then for any \(x \in M\), \(\nu ^{cu}\) is absolutely continuous with respect to Lebesgue measure on the weak unstable leaf \(\mathcal {W}_\Phi ^{cu}(x)\). Furthermore, the density \(e^{\psi }\) is Hölder continuous and smooth within the leaf \(\mathcal {W}_\Phi ^{cu}(x)\), and

where \(J^u_y(t)\) is the Jacobian of the map \(D \Phi ^t|_{E_\Phi ^u(y)}{:}\, E_\Phi ^u(y) \rightarrow E_\Phi ^u(\Phi ^t(y))\).

### Proof

Fix \(x \in M\) and choose a neighbourhood \(O(x)=U(x)\times V(x)\) with local product structure and let \(\varphi \) be coordinates on *O*(*x*) as described above. Fix a point \(p=\varphi (y,z)\in O(x)\). On the one hand, by the construction recalled previously, we have

On the other hand, the measure \(\mu \) has local product structure, hence there exists a positive Borel function \(\rho {:}\,U(x)\times V(x)\) such that

where \(\{\mu ^{cu}_q\}_{q \in O(x)}\), resp. \(\{\mu ^{s}_q\}_{q \in O(x)}\) is a system of conditional measures of \(\mu \) for the foliation \(\mathcal {W}_\Phi ^{cu}\), resp. \(\mathcal {W}_\Phi ^{s}\). As the foliations \(\mathcal {W}_\Phi ^{cu}\) and \(\mathcal {W}_\Phi ^{s}\) are absolutely continuous, for almost every \(q \in O(x)\), the conditional measure \(\mu ^{cu}_q\), resp. \(\mu ^{s}_q\) is absolutely continuous with respect to the Lebesgue measure on the leaf \(\mathcal {W}_\Phi ^{cu}(q)\), resp. \(\mathcal {W}_\Phi ^{s}(q)\). If we fix \(z \in V(x)\), we deduce from (A.3) to (A.4) and the previous discussion that

where *dy* denotes the Lebesgue measure on \(U(x)\subset \mathcal {W}_\Phi ^{s}(x)\). Applying \((\Phi ^t)_*\), it follows from (A.1) and (A.5) that

for some measurable function \(\psi {:}\,M \rightarrow {\mathbb {R}}\), where \(J^u_y(t)\) is the unstable Jacobian of \(D \Phi ^t\) at *y*. Thus, for almost every *y*, it holds

In other words, \(\psi \) is a measurable transfer function making \(\log J^u\) cohomologous to a constant. By Livsic’s theorem, \(\psi \) coincides almost everywhere with a Hölder solution which is smooth along the unstable leaves and Hölder transversally. With the upgraded regularity, we define an a priori new family of conditionals along the unstable leaves at every point. By uniqueness of the family of measures satisfying (A.1) (up to multiplicative constant), these must coincide with \(\nu ^{cu}\) up to fixed scalar. Therefore, \(\nu ^{cu}\) is absolutely continuous with smooth density. \(\square \)

### Remark A.2

By Proposition A.1, for any periodic point *y* of period \( \mathcal {L}(y)>0\), taking \(t=\mathcal {L}(y)\) in (A.2), we obtain another proof of Proposition 3.1 when \(\Phi \) is topologically mixing. Conversely, if (3.1) holds for any periodic orbit, then by Livsic’s theorem, the \(C^1\) cocycle

over \(\Phi \) is a coboundary, and (A.2) follows.

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De Simoi, J., Leguil, M., Vinhage, K. *et al.* Entropy rigidity for 3D conservative Anosov flows and dispersing billiards.
*Geom. Funct. Anal.* **30**, 1337–1369 (2020). https://doi.org/10.1007/s00039-020-00547-z

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DOI: https://doi.org/10.1007/s00039-020-00547-z