An entropic characterization of the flat metrics on the two torus
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Abstract
The geodesic flow of the flat metric on a torus is minimizing the polynomial entropy among all geodesic flows on this torus. We prove here that this properties characterises the flat metric on the two torus.
Keywords
Polynomail entropy Rigidity Weak KAM theory Geodesic flowRésumé
Le flot géodésique des métriques plates sur un tore minimise l’entropie polynomiale parmi tous les flots géodésique sur ce tore. On montre ici que cette propriété caractérise les métriques plates en dimension deux.
Mathematics Subject Classification
53D25 53C22 37B401 Introduction
There are several classes of hyperbolic manifolds on which the metrics with constant curvature are characterized by the fact that their geodesic flow is minimizing the topological entropy, see [6, 16] for example. The situation is different on tori. Flat metrics have zero entropy, but other metrics also have zero entropy, such as the tori of revolution. In order to characterize the flat metrics, it is therefore useful to consider a finer dynamical invariant of the geodesic flow, such as the polynomial entropy, introduced in [22].
Using the techniques of [22], it was proved in [17] that the polynomial entropy of a flat torus of dimension d (in restriction to the sphere bundle) is equal to \(d1\), which is a lower bound for the polynomial entropy of all metrics on \(\mathbb {T}^d\). It was also proved in [19] that the polynomial entropy of the revolution two torus is two, which is higher than the one of the flat two tori. This gives an indication that the polynomial entropy might be a sufficiently fine invariant to characterize the flat metrics. Our main result in the present paper is that this is indeed the case in dimension two. A partial result in that direction has been obtained in [18].
Theorem 1
If the polynomial entropy of a \(C^2\) metric g on \(\mathbb {T}^2\) (in restriction to the sphere bundle) is smaller than two, then this entropy is equal to one and the torus \((\mathbb {T}^2, g)\) is isometric to a flat torus.
Theorem 1 immediately implies the following strong rigidity for flat metrics on \(\mathbb {T}^2\) (see Corke and Kleiner [9]):
Corollary 1.1
Let g be a flat metric on \(\mathbb {T}^2\), and \(g'\) be another metric on \(\mathbb {T}^2\). If the geodesics flows of g and \(g'\) are \(C^0\) conjugated, then g and \(g'\) are isometric.
The detailed proof of Corollary 1.1 from Theorem 1 is given in Sect. 3.2 below. We will prove Theorem 1 using MatherFathi theory. The useful facts from this theory are recalled in Sect. 3, where a more general estimate on the polynomial entropy of Tonelli Hamiltonians is given, see Theorem 2. Theorem 1 is deduced from Theorem 2 using the Theorem of Hopf and its variants, see [15]. The definition of the polynomial entropy is recalled in Sect. 2, and the entropy estimates leading to the proof of Theorem 2 are detailed in Sect. 4. Once the dynamics has been well understood with the help of MatherFathy theory, these estimates are similar to those appearing in [19, 21, 22].
2 The polynomial entropy
Definition 2.1
We also consider sets that are \(\varepsilon \) separated for the metrics \(d_n^f\) (we will write \((n,\varepsilon )\)separated). Recall that a set E is said to be \(\varepsilon \)separated for a metric d if for all (x, y) in \(E^2\), \(d(x,y)\ge \varepsilon \). Denote by \(S_n^f(\varepsilon )\) the maximal cardinal of a \((n,\varepsilon )\)separeted set contained in X.
Remark 2.1
The following properties of the polynomial entropy are proved in [22].
Property 2.1
 1.
\(\mathrm{{h_{pol}}}\) is a \(C^0\) conjugacy invariant, and does not depend on the choice of topologically equivalent metrics on X.
 2.
If A is an finvariant subset of X, then \(\mathrm{{h_{pol}}}(f_{\vert _A})\leqslant \mathrm{{h_{pol}}}(f)\).
 3.
For \(m\in \mathbb N^*\), \(\mathrm{{h_{pol}}}(f^m) =\mathrm{{h_{pol}}}(f)\) and if f is invertible, \(\mathrm{{h_{pol}}}(f^{m})=\mathrm{{h_{pol}}}(f)\).
We conclude this section with the following useful result which relates the polynomial entropy of a flow with that of a Poincaré map.
Proposition 2.1

for any \(a\in A\), there exists \(t>0\) such that \(\phi ^{t}(a)\in \Sigma \).

for any \(a\in A\cap \Sigma \), X(a) is transverse to \(\Sigma \).
Proof
Let \(\tau :A\cap \Sigma \rightarrow \mathbb R^*_+ :a \mapsto \tau _a\) be the first return time map of \(\varphi \).
Since the function \(\tau \) is continuous on the compact set \(A\cap \Sigma \), we have \(T:=\max \{\tau _a\,\, a\in A\cap \Sigma \}<\infty \). Let \(d_\Sigma \) be the distance induced by d on \(\Sigma \).
We will now prove that two points x and y of \(A\cap \Sigma \) which are \((n,\varepsilon )\)separated by \(\varphi \) are \((nT,\delta \varepsilon )\) separated by \(\phi \) for \(\varepsilon <\tau ^*\). There exists \(m\in \{0,\dots ,n\}\) such that \(d_\Sigma (\varphi ^m(x),\varphi ^m(y))\geqslant \varepsilon \). Let us assume for definiteness that \(\tau ^m_x\leqslant \tau ^m_y\).
If \(\phi ^{\tau _x^m}(y)\in A{\setminus }K\), then \(d(\phi ^{\tau ^m_x}(x),\phi ^{\tau ^m_x}(y))\geqslant \delta \varepsilon \) by (2), hence x and y are \((\tau ^m_x,\delta \varepsilon )\)separated by \(\phi \).
If \(\phi ^{\tau _x^m}(y)\in K\), then there exists \(m'\leqslant m\) and \(s\in [\tau ^*,\tau ^*]\) such that \(\phi ^{\tau ^m_x}(y)=\phi ^s(\varphi ^{m'}(y))\).
If \(m'=m\), then \(d(\phi ^{\tau ^m_x}(x),\phi ^{\tau ^m_x}(y))=d(\varphi ^m(x),\phi ^{s}(\varphi ^m(y))\geqslant \delta \max (s,\varepsilon )\), hence x and y are \((\tau ^m_x,\delta \varepsilon )\)separated by \(\phi \).
If \(m'<m\), since \(\tau _x^{k+1}\tau _x^k>2\tau ^*\) for \(k\in \{1,\dots ,m1\}\), there exists \(k\in \{1,\dots ,m\}\) such that \(\phi ^{\tau _x^k}(y)\notin K\), otherwise \(m'\geqslant m\). Then \(d(\phi ^{\tau ^k_x}(x), \phi ^{\tau ^k_x}(y))\geqslant \delta \varepsilon \) by (2), hence the points x and y are \((\tau ^k_x,\delta \varepsilon )\)separated by \(\phi \).
3 Tonelli Hamiltonians
3.1 Some definitions from weak KAM theory
There is an interesting relation between the function \(c\longmapsto \alpha (c)\) and the rotation numbers of Mather measures, which was discovered by Mather [26]: The function \(\alpha \) is convex and superlinear on \(H^1(\mathbf{T },\mathbb R)\). Moreover, its subdifferential \(\partial \alpha (c)\in H_1(\mathbf{T },\mathbb {R})\) in the sense of convex analysis is precisely the set of rotation numbers of Mather measures at cohomology c.
Following Mather, we denote by \(\beta (h):H_1(\mathbf{T },\mathbb {R})\longrightarrow \mathbb {R}\) the Legendre dual of \(\alpha \). In the geodesic case, where H is quadratic in the fibers, the functions \(\alpha \) and \(\beta \) are homogeneous of degree 2. The function \(\sqrt{\beta }\), which is homogeneous of degree one, is called the stable norm.
3.2 Flat metrics and proof of the Corollary 1.1
Proposition 3.1
If the Hamiltonian flows of two Hamiltonians \(H(q,p)=\frac{1}{2}\langle G^{1}p,p\rangle \) and \(\tilde{H}(q,p)=\frac{1}{2}\langle \tilde{G}^{1}p,p\rangle \) associated to constant metrics are topologically conjugated (in restriction to their energy levels \(\{H=1\}\) and \(\{\tilde{H}=1\}\)), then there exists a matrix \(A\in Gl_d(\mathbb {Z})\) such that \( G =A^t \tilde{G} A\). As a consequence, the corresponding metrics are isometric.
Proof
Let us consider the quadratic functions \(n(v)=\langle Gv,v\rangle /2\) and \(\tilde{n}(v)=\langle \tilde{G}v,v\rangle /2\) on \(\mathbb {R}^d\) and let us denote \(\Sigma \) and \(\tilde{\Sigma }\) their unit spheres \(\Sigma =\{n=1\}\). The map \((q,p)\longmapsto (q, G^{1}p)\) conjugates the Hamiltonian flow of H in restriction to the energy level \(H=1\) to the flow of the equations \(\dot{q}=v, \dot{v}=0\) on \(\mathbf{T }\times \Sigma \). Since the flows of H and \(\tilde{H}\) are conjugated, there exists a homeomorphism \(\varphi : \mathbf{T }\times \Sigma \longrightarrow \mathbf{T }\times \tilde{\Sigma }\) which conjugates the flows of \(\dot{q}=v, \dot{v}=0\) on these manifolds.
Proof of Corollary 1.1
If the geodesic flow of the metric \(g'\) is topologically conjugated to the geodesic flow of the flat metric g, then the polynomial entropy of \(g'\) is equal to the polynomial entropy of g, which is equal to 1. Theorem 1 implies that \(g'\) is flat, and Proposition 3.1 then implies that g and \(g'\) are isometric. \(\square \)
3.3 The special case of dimension two, the main statement in the Tonelli case
In this section, we work on the twodimensional torus \(\mathbf{T }=\mathbb {R}^2/\mathbb {Z}^2\). We first recall from [12] some useful facts on rotation sets of flows on \(\mathbf{T }\).

If the straight line has rational direction [which means that it contains an element of \(H_1(\mathbf{T },\mathbb {Z})\)], then the ergodic invariant measures of nonzero rotation number are supported on periodic orbits. The \(\alpha \) and \(\omega \) limit sets of the flow are made of periodic orbits in this case.

If the straight line has irrational direction, then there is at most one ergodic invariant measure of nonzero rotation number.

If \(\rho (c)\) has rational direction, then the ergodic cminimizing measures are supported on periodic orbits. Moreover, the \(\alpha \) and \(\omega \) limits of each orbit of \(\mathcal {A}^*(c)\) are periodic orbits supporting cminimizing measures.

If \(\rho (c)\) has irrational direction, then there exists a unique cminimizing measure, and the rotation set \(\partial \alpha (c)\) is a point.
Let us explain a bit more how different rotation numbers can appear in the rational case. In this case, the periodic orbits of \(\mathcal {A}(c)\) are oriented embedded closed curve, and they all represent the same homology class \([\rho (c)]\in H_1(\mathbf{T },\mathbb {Z})\) which is the only indivisible integer class in the half line \(\rho (c)\). The rotation number of the invariant measure supported on such an orbit is then \([\rho (c)]/T\), where T is the minimal period of the orbit. The periodic orbits of \(\mathcal {A}(c)\) do not necessarily all have the same period, hence the associated measures do not necessarily have the same rotation number.
Proposition 3.2
 1.
\(\rho (c)\) has irrational direction and \(F(c)=\{c\}\).
 2.
\(\rho (c)\) has rational direction, \(\mathcal {M}(c)=\mathbf{T }\), and \(F(c)=\{c\}\)
 3.
\(\rho (c)\) has rational direction, \(\mathcal {M}(c)\ne \mathbf{T }\) and F(c) is a nontrivial segment \([c^,c^+]\). The sets \(\mathcal {A}^*(c^)\) and \(\mathcal {A}^*(c^+)\) contain nonperiodic orbits (which are heteroclinics).
If, for a given value \(e>\min \alpha \) of the energy, case 3 does not occur for any \(c\in \alpha ^{1}(e)\), then the map \(c\longmapsto \rho (c)\) is a homeomorphism from \(\alpha ^{1}(e)\) to \(SH_1(\mathbf{T },\mathbb {R})\). The energy level \(\{H=e\}\) is then \(C^0\)integrable, as is proved in ([24], Theorem 3), see also Sect. 3.4:
Proposition 3.3
If, for a given value \(e>\min \alpha \) of the energy, case 3 does not occur for any \(c\in \alpha ^{1}(e)\), then the Aubry sets \(\mathcal {A}^*(c), c\in \alpha ^{1}(e)\) are Lipschitz invariant graphs which partition the energy level \(\{H=e\}\).
If H is the Hamiltonian associated to a Riemaniann metric, then this implies that the metric is flat, in view of the Theorem of Hopf, see also [15]. As a consequence, Theorem 1 follows from:
Theorem 2
Let \(e>\min \alpha \) be a given energy level. If there exists a cohomology \(c\in \alpha ^{1}(e)\) in case 3, then the polynomial entropy of the Hamiltonian flow restricted to the energy level \(\{H=e\}\) is not less than 2. In other words, if the polynomial entropy of the flow restricted to the energy level \(\{H=e\}\) is less than two, the Aubry sets \(\mathcal {A}^*(c), c\in \alpha ^{1}(e)\) are Lipschitz invariant graphs which partition the energy level.
We will make use in the proof of two important properties of the Aubry sets:
Property 3.1
The setvaled map \(c\longmapsto \mathcal {A}^*(c)\) is outer semicontinuous. It means that each open set \(U\subset T^*\mathbf{T }\) containing \(\mathcal {A}^*(c)\), also contains \(\mathcal {A}^*(c')\) for \(c'\) close to c.
We recall the definition of the vectorfield \(\chi (q):=\partial _pH(q,c+dw(q))\) on \(\mathbf{T }\).
Property 3.2
For each \(c\in \alpha ^{1}(e)\), there exists a global curve of section of \(\mathcal {A}(c)\). More precisely, there exists a cooriented \(C^1\) embedded circle \(\Sigma \subset \mathbf{T }\) such that \(\chi (q)\) is transverse to \(\Sigma \) on \(\mathcal {A}(c)\) and respects the coorientation. Moreover, each half orbit of \(\mathcal {A}(c)\) intersects \(\Sigma \). The flow of \(\mathcal {A}(c)\) thus induces a homeomorphism \(\psi \) of \(\Sigma \cap \mathcal {A}(c)\) which preserves the cyclic order of \(\Sigma \).
Proof of Property 3.2
3.4 Faces of the balls of \(\alpha \) on the two torus
We take \(d=2\) and fix an energy level \(e>\min \alpha \). We study the affine parts of the ball \(\alpha ^{1}(e)\) and prove Propositions 3.2 and 3.3. The following is a variant of a Lemma of Massart [23]:
Lemma 3.1
Assume that \(d=2\), that \(e>\min \alpha \), and that the segment \([c_0,c_1]\) is contained in \(\alpha ^{1}(e)\). Then the Mather set \(\mathcal {M}^*(c)\) does not change when c varies in \([c_0,c_1]\).
Proof
The following Lemma also comes from Massart [23]:
Lemma 3.2
Assume that \(e>\min \alpha \), and that the segment \([c_0,c_1]\) is contained in \(\alpha ^{1}(e)\). Then the Aubry set \(\mathcal {A}^*(c)\) does not change when c varies in \(]c_0,c_1[\). Moreover, we have the inclusion \(\mathcal {A}^*(c)\subset \mathcal {A}^*(c_0) \cap \mathcal {A}^*(c_1)\) for each \(c\in \, ]c_0,c_1[\).
We will see that, unlike the Mather set, the Aubry set can be bigger at the boundary points.
Proof
Let us consider a point \(c=ac_0+(1a)c_1, a\in \, ]0,1[\). Let \(w_i\), \({i\in \{0,1\}}\) be a \(c_i\)critical subsolution strict outside the Aubry set. Then, \(w_c:=aw_0+(1a)w_1\) is a ccritical subsolution. Using the strict convexity of H in p, we observe that the strict inequality \(H(q,c+sw_c(q))<e\) holds outside of the set where \(H(q,c_0+dw_0(q))=e\) and \(H(q,c_1+dw_1(q))=e\) and \(c_0\,+\,dw_0= c_1+dw_1\). We conclude that the Aubry set \(\mathcal {A}(c)\) is contained in \(\mathcal {A}(c_0)\cap \mathcal {A}(c_1)\), and that \(c_0+dw_0=c_1+dw_1=c+dw_c\) on \(\mathcal {A}(c)\). As a consequence, \(\mathcal {A}^*(c)\subset \mathcal {A}^*(c_0)\cap \mathcal {A}^*(c_1)\). If c and \(c'\) are two points in \(]c_0,c_1[\), assuming for example that \(c\in \,]c_0,c'[\), we conclude that \(\mathcal {A}^*(c)\subset \mathcal {A}(c')\). Similarly, we have \(c'\in \, ]c,c_1[\), and we obtain the converse inclusion, hence \(\mathcal {A}^*(c)=\mathcal {A}^*(c')\). \(\square \)
We recall that F(c) is defined as the largest segment of \(\alpha ^{1}(e)\) containing c.
Corollary 3.1
If \(\mathcal {M}(c)=\mathbf{T }\), then \(F(c)=c\).
Proof
If \(\mathcal {M}(c)=\mathbf{T }\), then there exist one and only one ccritical subolution w, and \(\mathcal {M}^*(c)\) is the graph of \(c+dw\). Assume now that there exists \(c'\) such that \(\mathcal {M}^*(c')=\mathcal {M}^*(c)\). Then \(\mathcal {M}(c')=\mathbf{T }\), hence there exists a unique \(c'\)critical subsolution \(w'\), and \(\mathcal {M}^*(c')\) is the graph of \(c'+dw'\). We thus have \(c+dw=c'+dw'\) at each point, which implies that \(c=c'\). In view of Lemma 3.1, this implies that \(F(c)=c\). \(\square \)
The following was first proved by Bangert, see [2]:
Corollary 3.2
If \(\rho (c)\) has an irrational direction, then \(F(c)=c\).
Proof
As above, let us consider a cohomology c satisfying the hypothesis of the Corollary, and a cohomology \(c'\) such that \([c,c']\in \alpha ^{1}(e)\), hence \(\mathcal {M}^*(c')=\mathcal {M}^*(c)\), by Lemma 3.1. We will prove that \(c'=c\), which implies the Corollary. Note that \((c'c)\cdot \rho (c)=0\), so that it is enough to prove that \((c'c)\cdot [\Sigma ]=0\), where \([\Sigma ]\in H_1(\mathbf{T },\mathbb {Z})\) is the homology of the section \(\Sigma \) given by Property 3.2 (equipped with an orientation). Let w and \(w'\) be c and \(c'\)critical subsolutions. We consider the closed Lipschitz form \(\eta =c'c+dw'dw\), whose cohomology is \(c'c\) and prove that \(\int _{\Sigma }\eta =0\).
Since the intervals \(\psi ^k(I)\) are two by two disjoint in \(\Sigma \), their lengh is converging to zero. Since the form \(\eta \) is bounded, this implies that \(\int _{\psi ^k(I)}\eta \longrightarrow 0\), hence that \(\int _I \eta =0\). \(\square \)
We consider the cooriented section \(\Sigma \) given by Property 3.2. We orient \(\Sigma \) in such a way that \((c^+c^)\cdot [\Sigma ]\geqslant 0\), where \([\Sigma ]\) is the homology of \(\Sigma \) (hence \((c^+c^)\cdot [\Sigma ]> 0\) if \(c^+\ne c ^\), since \([\Sigma ]\) is not proportional to \(\rho (c)\)). The return map \(\psi \) from \(\mathcal {M}(c)\cap \Sigma \) to istelf is periodic, its minimal period is \(\tau :=\sigma \cdot [\rho (c)]\). The complement of \(\mathcal {M}(c)\) in \(\mathbf{T }\) is a disjoint union of topological open annuli. Each of these annuli U intersects \(\Sigma \) in \(\tau \) disjoint open intervals, that we orient according to the orientation of \(\Sigma \). Each orbit of \(\mathcal {A}(c)\mathcal {M}(c)\) is contained in such an annulus U, is \(\alpha \)asymptotic to one of its boundaries, and is \(\omega \)asymptotic to its other boundary. We say that such an orbit is positive if it crosses the annulus U according to the orientation of \(\Sigma \), and that it is negative if it crosses in the other direction. In other words, the heteroclinic orbit is positive if the sequence of its successive intersections with the interval I is increasing. The following implies Proposition 3.2:
Proposition 3.4

\(c^\ne c^+\)

The Aubry set \(\mathcal {A}(c^+)\) contains positive heteroclinics in each connected component of \(\mathbf{T }\mathcal {M}(c)\) and no other orbit except those of \(\mathcal {M}(c)\).

The Aubry set \(\mathcal {A}(c^)\) contains negative heteroclinics in each connected component of \(\mathbf{T }\mathcal {M}(c)\) and no other orbit except those of \(\mathcal {M}(c)\).

Finally, \(\mathcal {A}(c)=\mathcal {M}(c)\) for each c in \(]c^,c^+[\).
Proof
The statements concerning \(\mathcal {A}(c^+)\) and \(\mathcal {A}(c^)\) imply that \(c^+\ne c^\). Moreover, they imply that \(\mathcal {A}(c^+)\cap \mathcal {A}(c^)=\mathcal {M}(c)\), hence that \( \mathcal {A}(c)=\mathcal {M}(c)\) for each c in \(]c^,c^+[\), by Lemma 3.2.
We will now prove the statement concerning \(\mathcal {A}(c^+)\), the other one being similar. We fix a connected component U of \(\mathbf{T }\mathcal {M}(c)\), and a connected component I of \(U\cap \Sigma \). Let \(\rho _m\) be the direction of \(m[\rho ]+[\Sigma ]\), and let \(c_m\in \alpha ^{1}(e)\) be such that \(\rho (c_m)=\rho _m\). Note that \(c_m\longrightarrow c^+\).
The annulus U contains an oriented closed curve K of homology \([\rho ]\). The Aubry set \(\mathcal {A}(c_m)\) contains a periodic orbit of homology positively proportional to \(m[\rho ]+[\Sigma ]\), hence it intersects K. In view of the semicontinuity of the Aubry set, see Property 3.1, we deduce that \(\mathcal {A}(c^+)\) intersects K. As a consequence, the set \(\mathcal {A}(c^+)\) does contain heteroclinic orbits in U. Moreover, there exists \(m_0\in \mathbb {N}\) and a compact subinterval \(J\subset I\) which contains a point in each orbit of \(\mathcal {M}(c_m)\) for \(m\geqslant m_0\).
Let us now consider the return map \(\psi ^{\tau }\) of \(I\cap \mathcal {A}(c)\). Either we have \(\psi ^{\tau }(x)> x\) for each \(x\in I\cap \mathcal {A}(c)\), and the orbits of \(\mathcal {A}(c)\cap U\) are positive heteroclinics, or we have \(\psi ^{\tau }(x)< x\) for each \(x\in I\cap \mathcal {A}(c)\), and the orbits of \(\mathcal {A}(c)\cap U\) are negative heteroclinics.
In view of the semicontinuity of the Aubry set, \(\Sigma \) is also a cooriented transverse section for \(\mathcal {A}(c_m)\) for \(m\geqslant m_0\), provided \(m_0\) is large enough. Denoting by \(\psi _{c_m}\) the corresponding section map, we have \(\psi ^{\tau }_{c_m}(J)\subset I\) for \(m\geqslant m_0\) provided \(m_0\) is large enough. Then, there exists a point \(x_m\in J\cap \mathcal {M}(c_m)\subset J\cap \mathcal {A}(c_m)\), and \(x_m<\psi ^{\tau }_{c_m}(x_m)\).
At the limit, using the semicontinuity of the Aubry set, we find a point \(x\in J\cap \mathcal {A}(c^+)\) such that \(\psi ^{\tau }(x)\geqslant x\), hence \(\psi ^\tau (x)>x\). We conclude that the heteroclinics are positive. \(\square \)
For the convenience of the reader, and because our statement is not exactly the one of [24], Theorem 3, we now prove Proposition 3.3, following [24]:
We consider an energy level \(e> \min \alpha \) such that the curve \(\alpha ^{1}(e)\) does not contain any nontrivial segment, which is equivalent to saying that \(\mathcal {M}(c)=\mathbf{T }\) for each c such that \(\rho (c)\) is rational. Note then that the map \(\rho : \alpha ^{1}(e)\longrightarrow SH_1(\mathbf{T },\mathbb {R})\) is continuous and bijective, hence it is a homeomorphism. Since the set \(SH_1(\mathbf{T },\mathbb {Z})\) of rational directions is dense in \(SH_1(\mathbf{T },\mathbb {R})\), its preimage \(\rho ^{1}(SH_1(\mathbf{T },\mathbb {Z}))\) is dense in \(\alpha ^{1}(e)\). For each point c in this set, we have \(\mathcal {A}(c)=\mathbf{T }\). In view of the semicontinuity of the Aubry set, we deduce that \(\mathcal {A}(c)=\mathbf{T }\) for each \(c\in \alpha ^{1}(e)\). As a consequence, there exists a unique (up to the addition of a constant) ccritical subsolution \(w_c\), which is actually a solution, and the Aubry set \(\mathcal {A}^*(c)\) is the graph of \(c+dw_c\). Moreover, the functions \(dw_c, c\in \alpha ^{1}(e)\) are equiLipschitz. The semicontinuity of the Aubry set \(\mathcal {A}^*\) implies that the map \(c\longmapsto c+dw_c(q)\) is continuous for each \(q\in \mathbf{T }\).
The orbits of \(\mathcal {A}^*(c)\) all have a forward rotation number in \(\rho (c)\). For \(c'\ne c\), the orbits of \(\mathcal {A}^*(c')\) all have a forward rotation number in \(\rho (c')\), and, since \(\rho (c')\ne \rho (c)\), the sets \(\mathcal {A}^*(c)\) and \(\mathcal {A}^*(c')\) are disjoint. As a consequence, for each \(q\in \mathbf{T }\), the map \(c\longmapsto c+dw_c(q)\) is one to one on \(\alpha ^{1}(e)\), hence it has degree \(\pm 1\) as a circle map into \(\{p\in T_q\mathbf{T }: H(q,p)=e\}\). It is thus onto, which implies that the Aubry sets fill the energy level. \(\square \)
4 Lower bound for the polynomial entropy
We prove Theorem 2. We consider an energy level \(e>\min \alpha \), assume that the ball \(\alpha ^{1}(e)\) contains a nontrivial face \([c^,c^+]\), and prove that the entropy of the Hamiltonian flow on the energy level \(H^{1}(e)\) is at least two. The proof have similarities with the ones of [19, 21, 22]. We work with the section \(\Sigma \) of \(\mathcal {A}(c^+)\) given in Property 3.2. We fix a parameterisation \(\mathbb {R}/\mathbb {Z}\longrightarrow \Sigma \), and put on \(\Sigma \) the distance such that this parameterisation is isometric. This distance is Lipschitz equivalent to the restriction of the distance on \(\mathbf{T }\).
The direction \(\rho (c)\in SH_1(\mathbf{T },\mathbb {Z})\) is independant of \(c\in [c^,c^+]\), and it is rational, we denote it by \(\rho \) in the sequel, and by \([\rho ]\in H_1(\mathbf{T },\mathbb {Z})\) the associated indivisible integer point. As above, we consider, for \(m\in \mathbb {N}\), the direction \(\rho _m\) of \(m[\rho ]+[\Sigma ]\) and a cohomology \(c_m\in \alpha ^{1}(e)\) such that \(\rho (c_m)=\rho _m\). Let us decide for definiteness that \(c_m\longrightarrow c^+\) (otherwise we exchange the names of \(c^+\) and \(c^\)). We fix \(m_0\) large enough so that \(\Sigma \) is a cooriented transverse section of \(\mathcal {A}(c_m)\) for each \(m\geqslant m_0\) (such a value of \(m_0\) exists in view of the semicontinuity of the Aubry set) and denote by \(\psi _{c_m}\) the corresponding return map of \(\mathcal {A}(c_m)\cap \Sigma \). The orbits of \(\mathcal {M}(c)\) give rise to periodic orbits of \(\psi \), and the minimal period of these orbits is \(\tau := \sigma \cdot [\rho ]\), where \(\sigma \) is the cohomology of the intersection cocycle associated to \(\Sigma \). The integral class \(m[\rho ]+[\Sigma ]\) is not necessarily indivisible in \(H_1(\mathbf{T },\mathbb {Z})\), hence the minimal period for the return map \(\psi _{c_m}\) of the points of \(\Sigma \cap \mathcal {M}(c_m)\) may be smaller than \(m\tau \). However, because \(m[\rho ]+[\Sigma ]\) is indivisible in the group generated by \([\Sigma ]\) and \([\rho ]\), we have:
Lemma 4.1
Each point of \(\mathcal {M}(c_m)\cap \Sigma \) is periodic for the map \(\psi ^{\tau }_{c_m}\), and has a minimal period equal to m.
The following proof might appear unnecessarily complicated. Things can also be understood as follows: The statement of the Lemma is obvious if (\([\Sigma ]\),\([\rho ])\) is a base of \(H^1(\mathbf{T },\mathbb {Z})\) (since \(m[\rho ]+[\Sigma ]\) is then indivisible in \(H^1(\mathbf{T },\mathbb {Z})\)) and we can reduce the situation to this simple case by taking a finite covering, which does not change the value of the polynomial entropy.
Proof
The minimal \(\psi _{c_m}^{\tau }\)period of orbits of \(\mathcal {M}(c_m)\cap \Sigma \) is equal to the intersection number \(\tilde{\sigma }\cdot [\tilde{\rho }_m]=(\sigma /\tau ) \cdot (m[\rho ]+[\Sigma ])=m\). \(\square \)
Lemma 4.2
Proof
Let \(\theta \) and \(\theta '\) be two points of this orbit. There exists \(l\in \{0,1,\ldots ,m1\}\) such that \(\psi ^{l\tau }(\theta )\in J\). Then \(\psi ^{l\tau }(\theta ')\) is another point of the same orbit, hence \(d\big (\psi ^{l\tau }(\theta ),\psi ^{l\tau }(\theta ')\big )\geqslant 2\epsilon _0\). \(\square \)
Lemma 4.3
Since the cardinal of this union is more than \(m^2\), we conclude that the polynomial entropy of \(\Psi \) is at least two. By Proposition 2.1 the polynomial entropy of the Hamiltoinan flow on the energy surface is at least two.
Proof
Let \(x=(q,p)\in O_{\Psi ^{\tau }}(x_k)\) and \(y=(\theta , \eta )\in O_{\Psi ^{\tau }}(x_l)\) be two different points in this union.
If \(k=l\), the points x and y belong to the same orbit \(O_{\Psi ^{\tau }}(x_k)\). They are \((\epsilon _0,m)\)separated by \(\Psi ^{\tau }\) in view of Lemma 4.2.
Otherwise, we assume for definiteness that \(m\leqslant k<l< 2m\). There exists an integer \(s\in \{0,1,\ldots , k1\}\) such that \(\psi ^{s\tau }_{c_k}(q)\in J\).
If \(d\big (\psi ^{s\tau }_{c_k}(q),\psi ^{s\tau }_{c_l}(q)\big )\geqslant \epsilon _0\), then \(d\big (\Psi ^{s\tau }(x), \Psi ^{s\tau }(y)\big )\geqslant \epsilon _0\) hence x and y are \((\epsilon _0, k)\)separated by \(\Psi ^{\tau }\).
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