Generic Hamiltonian Dynamics

Abstract

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a $C^2$-residual ${\mathcal {R}}$ in the set of $C^2$ Hamiltonians on a closed symplectic manifold $M$, such that, for any $H\in {\mathcal {R}}$, there is a full measure subset of energies $e$ in $H(M)$ such that the Hamiltonian level $(H,e)$ is topologically mixing; moreover these level sets are homoclinic classes.

Appendix: Hamiltonian Connecting Lemma for Pseudo-Orbits

This Appendix is quite technical and a more or less direct application of the arguments in [5] with the conceptual adaptations and perturbations required for Hamiltonians. In what follows, and for the sake of completeness, we intend to give a short description of the proof explaining the essential steps and the main differences in our setting. We notice that we do not claim any novelty in the following approach. The ideas follows [5] but the tools are, of course, with a Hamiltonian flavor.

As explained in [5, 7, 10], the proof of the Connecting Lemma for pseudo-orbits is divided in three main parts. The first step is to prove that the Connecting Lemma concerns on local perturbations. These perturbations motivate the definition of perturbation boxes whose support must be in the interior of small open sets, pairwise disjoint till a sufficiently large number of iterates. Separately, we need to analyze the dynamics near closed orbits with small period because these orbits are not contained in any perturbation box. Finally, we must analyze the global dynamics, in order to cover any orbit with perturbation flowboxes.

1.1 Local Perturbations

We proceed to describe the modifications of the local perturbation methods in the Hamiltonian context, namely the lift perturbation process and selection of tilings adapted to pseudo-orbits.

Lift axiom Fix $p\in Per(H)$ and a small neighborhood $U_p$ of $p$. By the Darboux Theorem (see, for example, [3, Theorem 3.2.2]), there is a smooth symplectic change of coordinates $\varphi _p:U_p\rightarrow T_{p}M$, such that $\varphi _p(p)=\vec {0}$. Denote by $N_{p,\delta }$ the ball centered in $\vec {0}$ at the normal fiber at $p$ and with radius $\delta$. For a given $\delta >0$ depending on $p$ we let $f_H: \varphi ^{-1}_{p}(N_{p,\delta })\rightarrow \varphi ^{-1}_{X^1_H(p)}(N_{X^1_H(p)})$ be the canonical Poincaré time-one arrival associated to $H$, in fact, given a regular point $p$, we can chose any $\tau >0$ less than its period, if $p$ is periodic. In [13], when proving the closing lemma for Hamiltonians, Pugh and Robinson show that the lift axiom is satisfied for Hamiltonians, and they obtain the closing from the lifting. In short, lifting is a way of pushing the orbit along a given direction by a small Hamiltonian perturbation $C^2$-close to the identity. We point out that we never have to push in the direction of increasing energies, i.e. it is possible to push only along the energy surface. Furthermore, we recall the key point on the using of the $C^1$ topology of the Hamiltonian vector field: “...one can lift points $p$ in prescribed directions $v$ with results proportional to the support radius” ([13, pp. 266]). Allowing that the perturbations can be done in several flowboxes the proportionality constant can be made arbitrarily close to one.

Lift Axiom for Hamiltonians (cf. [13], Sect. 9(a)]) Consider a Hamiltonian $H\in C^2(M,{\mathbb {R}})$ and let ${\mathcal {U}}$ be a $C^2$-neighborhood of $H$. Then there are $0<\epsilon \le 1$ and a continuous function $\delta :M\setminus Sing (X_H)\rightarrow (0,1)$, both depending on $H$ and on ${\mathcal {U}}$, such that, for any $p$ and $v\in N_{p,\delta (p)}\cap \varphi _p(H^{-1}(H(p)))$, there exists $\tilde{H}\in {\mathcal {U}}$ satisfying:

• $f^{-1}_H\circ f_{\tilde{H}}(p)=\varphi ^{-1}_p(\epsilon v)$;

• $supp(X_{\tilde{H}}-X_H)$ is contained in the flowbox $\mathcal {T}=\bigcup _{t\in (0,T)}X_H^t(B_{\Vert v\Vert }(p)),$ where $B_{\Vert v\Vert }(p)$ is taken in a transversal section of $p$ and $T=T(y)$ is such that $T(p)=1$ and $X_H^{T(y)}(y)\in B_{\Vert v\Vert }(X_H(p))$, for any $y\in B_{\Vert v\Vert }(p)$.

Tiled Sections and Perturbation Flowboxes Given a symplectic chart $\varphi :U\rightarrow {\mathbb {R}}^{2d}$, we say that the cross-section ${\mathcal {C}}$ to the flow on ${\mathcal {E}}_{H,e}$ on the chart $(U,\varphi )$ is a tiled section if $\varphi ({\mathcal {C}}) \subset {\mathbb {R}}^{2d}$ is symplectomorphic to the standard cube in ${\mathbb {R}}^{2d-2}$, tilled by smaller cubes by homotheties and translations (see [10], Fig. 1]). Write ${\mathcal {C}}=\displaystyle \cup _{k=1}^m \mathcal {T}_k$, with $m\in {\mathbb {N}}$, where each $\mathcal {T}_k$ is called a tile of ${\mathcal {C}}$.

Fig. 1

Representation of a pseudo-orbit preserving the tiling

Definition 2.1

Consider a Hamiltonian system $(H,e,{\mathcal {E}}_{H,e})$, a tiled section ${\mathcal {C}}=\displaystyle \cup _{k=1}^m \mathcal {T}_k$ on ${\mathcal {E}}_{H,e}$ and a constant $T>0$. We say the pseudo-orbit $\{x_i\}_{i=0}^n$ on ${\mathcal {E}}_{H,e}$ ($n\in {\mathbb {N}}$) preserves the tiling in the injective flowbox ${\mathcal {F}}_H({\mathcal {C}},T)=\displaystyle \cup _{t\in [0,T]}X^t_H({\mathcal {C}})$ if $x_0,x_n\notin {\mathcal {F}}_H({\mathcal {C}},T)$ and for any $i\in \{1,\ldots ,n-1\}$:

• if $X_H^{[0,1]}(x_i)\cap {\mathcal {C}}\in \mathcal {T}_k$ for some $k\in \{1,\ldots ,m\}$, then $X_H^{(-2,0)}(x_{i+1})\cap {\mathcal {C}}\in \mathcal {T}_k$ and

• if $X_H^1(x_i)\in X_H^{[1,T]}({\mathcal {C}})$, then $x_{i+1}\in {\fancyscript{O}}_H^+(x_i)$.

This definition implies that the intersection of the pseudo-orbit $\{x_j\}_{j=0}^n$ with the flowbox ${\mathcal {F}}_H({\mathcal {C}},T)$ is an union of segments of orbits such that the segment $X^{[0,1]}(x_i)$ intersects the tilled cross-section ${\mathcal {C}}$, and $x_{i+k}$ ($k\ge 1$) belong to the orbit of $x_{i+1}$ while the segment $X^{[0,1]}(x_{i+k})$ intersects the flowbox ${\mathcal {F}}_H({\mathcal {C}},T)$ (see Fig. 1). Moreover, as Pugh and Robinson explained in [13], Sect. 9(a)], local perturbations on $H$ do not change the energy hypersurfaces in the boundary of the flowboxes where the perturbations take place. So, we are allowed to push along energy levels.

The Hayashi Connecting Lemma is a key ingredient to prove the Connecting Lemma for pseudo-orbits of Hamiltonians and, as stated in [16], it can be adapted for Hamiltonians. In fact, following  [5], Théorème 5], we can extract a slightly stronger statement of the Connecting Lemma for Hamiltonians in [16, Theorem E], which can be seen as a theorem of existence of perturbation flowboxes:

Theorem 2.6

Given a Hamiltonian system $(H,e,{\mathcal {E}}_{H,e})$ and a $C^2$-neighborhood ${\mathcal {U}}$ of $H$, there exists $T>0$ such that if ${\mathcal {C}}$ is a tiled section, then ${\mathcal {F}}_H({\mathcal {C}},T)=\displaystyle \cup _{t\in [0,T]}X^t_H({\mathcal {C}})$ is a perturbation flowbox of length $T$, that is: for any pseudo-orbit $\{x_i\}_{i=0}^n$ on ${\mathcal {E}}_{H,e}$ preserving the tiling in ${\mathcal {F}}_H({\mathcal {C}},T)$, there exist $\tilde{H}\in {\mathcal {U}}$ preserving the energy hypersurface, such that $\tilde{H}=H$ restricted to the energy hypersurface and outside ${\mathcal {F}}_H({\mathcal {C}},T)$, and a pseudo-orbit $\{y_j\}_{j=0}^m$ on ${\mathcal {E}}_{\tilde{H},e}={\mathcal {E}}_{H,e}$, with $m\in {\mathbb {N}}$, such that $y_0=x_0$ and $y_m=x_n$ and the intersection of the pseudo-orbit $\{y_j\}_{j=0}^m$ with ${\mathcal {F}}_H({\mathcal {C}},T)$ is a segment of a true orbit of a point $y_j$ for $X^t_{\tilde{H}}$.

For notation simplicity we will call the set $supp({\mathcal {C}})=\displaystyle \cup _{t\in [0,T]}X^t_H(closure({\mathcal {C}}))$ the support of the perturbation flowbox ${\mathcal {C}}$ (inside ${\mathcal {E}}_{H,e}$).

1.2 Avoidable Closed Orbits and Covering Families

Notice that the jumps of a pseudo-orbit have no reason to respect the tiling of some perturbation flowbox and these are not definable for closed orbits with small period. To deal with this difficulty, we introduce the concept of avoidable closed orbits and of covering families.

Avoidable Closed Orbits This kind of orbits are used to derive perturbation flowboxes with disjoint supports, in such a way that the pseudo-orbits stay away from closed orbits with small period. We anticipate that, if ${\mathcal {E}}_{H,e}$ has no orbits with small period and all the closed orbits are uniformly avoidable, then we will be able to build a covering family of perturbation flowboxes for ${\mathcal {E}}_{H,e}$. The next definition is adapted from [5, Definition 3.10]. Consider a Hamiltonian system $(H,e,{\mathcal {E}}_{H,e})$ and a closed orbit $\gamma$ of $H$ on ${\mathcal {E}}_{H,e}$. Let ${\mathcal {U}}$ be a $C^2$-neighborhood of $H$ and fix $T>0$. We say a closed orbit $\gamma$ is avoidable for (${\mathcal {U}}, T$), if, for any neighborhood $V_0$ of $\gamma$ and for any $t>0$, there are $\epsilon >0$, open neighborhoods $W\subset V\subset V_0$ of the closed orbit $\gamma$ in ${\mathcal {E}}_{H,e}$, and a family of $\mathcal C =\{\mathcal C_i\}_i$ of tiled cross-sections so that the perturbation flowboxes ${\mathcal {F}}_H({\mathcal {C}}_i,T)=\displaystyle \cup _{t\in [0,T]}X^t_H({\mathcal {C}}_i)$ of length $T$ in ${\mathcal {E}}_{H,e}$ are contained in $V$ and have disjoint supports, and satisfies:

• (a) there exist two families of compacts ${\mathcal {I}}$ and ${\mathcal {O}}$ contained in the interior of the tiles of ${\mathcal {C}}$ such that any segment of any $\epsilon$-pseudo-orbit on ${\mathcal {E}}_{H,e}$ starting outside $V$ (respectively, inside $W$) and ending inside $W$ (respectively, outside $V$) intersects $X^{[0,1]}_H(I)$ for some compact $I\in {\mathcal {I}}$ (respectively, intersects $X^{[0,1]}_H(O)$ for some compact $O\in {\mathcal {O}}$);

• (b) for any compacts $I\in {\mathcal {I}}$ and $O\in {\mathcal {O}}$, there exist a pseudo-orbit on ${\mathcal {E}}_{H,e}$, with jumps inside $X^{[0,1]}_H({\mathcal {C}})$ and preserving the tile of ${\mathcal {C}}$, starting in $I$ and ending in $O$;

• (c) for any $x$ in the closure of ${\mathcal {F}}_H({\mathcal {C}},T)$, the first return time of $X^T_H(x)$ to the closure of the perturbation flowboxes is larger than $t$.

In a few words, a closed orbit $\gamma$ is avoidable for $({\mathcal {U}}, T)$ if, for any $t>0$, there exists a family of perturbation flowboxes of length $T$ (with tiled cross sections) such that, given a pseudo-orbit with starting and ending points far from $\gamma$, but passing very close of $\gamma$, we can exchange the segments of the pseudo-orbit passing close of $\gamma$ by segments of another pseudo-orbit with jumps inside the tiles replacing the original pseudo-orbit by another with smaller number of elements. By Theorem 1.2 the closed orbits of a $C^2$-generic Hamiltonian are uniformly avoidable.

The closed orbits for the Hamiltonian $H$ on ${\mathcal {E}}_{H,e}$ are called uniformly avoidable if they are isolated and there is a $C^2$-neighborhood ${\mathcal {U}}$ of $H$ and $T>0$ so that all closed orbits in ${\mathcal {E}}_{H,e}$ are avoidable for (${\mathcal {U}}, T$). Covering Families Given a Hamiltonian system $(H,e,{\mathcal {E}}_{H,e})$, we want to cover the orbits on ${\mathcal {E}}_{H,e}$ by a family of perturbation flowboxes, with pairwise disjoint supports. In general, if ${\mathcal {E}}_{H,e}$ contains closed orbits with small period, then ${\mathcal {E}}_{H,e}$ cannot have a covering family. In fact, this kind of closed orbits are disjoint from the perturbation flowboxes. This motivates the definition of covering families outside ${\mathcal {V}}=\cup _{j=1}^r V_j$, where the sets $V_j$ ($1\le j\le r$) are, in fact, neighborhoods of these closed orbits with small period.

Let ${\mathcal {U}}$ be a $C^2$-neighborhood of $H$ and let $({\mathcal {F}}_H({\mathcal {C}}_i,T))_i$ denote a family of perturbation flowboxes for ($H,{\mathcal {U}}$), with pairwise disjoint supports, and ${\mathcal {V}}$ denote a finite family of non-empty open subsets of ${\mathcal {E}}_{H,e}$ with pairwise disjoint supports. We say that a family $({\mathcal {F}}_H({\mathcal {C}}_i,T))_i$ of perturbation flowboxes for ($H,{\mathcal {U}}$) with disjoint supports is a covering family of ${\mathcal {E}}_{H,e}$ if there exists a family of compact subsets $\mathcal D \subset \bigcup _i interior({\mathcal {C}}_i)$ and $t>0$ so that any orbit segment of $x\in {\mathcal {E}}_{H,e}$ of length $\ge t$ intersects some element in $\mathcal D$. The following definition is an adaption of [5, Definition 3.2] for Hamiltonians. A finite set of perturbation flowboxes $({\mathcal {F}}_H({\mathcal {C}}_i,T))_i$ for $(H,{\mathcal {U}})$ and with pairwise disjoint supports is said to be a covering family of ${\mathcal {E}}_{H,e}$ outside ${\mathcal {V}}$ if there are

• $t>0$ and $\epsilon >0$;

• an open set $W_j$ and a compact set $F_j$, such that $F_j\subset W_j\subset V_j$, for every $j\in \{1,\ldots ,r\}$;

• a finite family of compacts $\mathcal {D}=\displaystyle \cup _{i=1}^s D_i$ on ${\mathcal {C}}$, such that every $D_i$ is contained in the interior of a tile of ${\mathcal {C}}=\bigcup _i {\mathcal {C}}_i$;

• two families ${\mathcal {I}}_{j}$ and ${\mathcal {O}}_{j}$ contained in $\mathcal {D}$ such that the support of the flowboxes of the tiles of ${\mathcal {C}}$ containing this compacts is contained in $V_j$, for any $j\in \{1,\ldots ,r\}$,

such that any segment of any $\epsilon$-pseudo-orbit on ${\mathcal {E}}_{H,e}$:

• (a) with length $\ge t$ intersects some $F_j$ or a compact of $\mathcal {D}$;

• (b) starting outside $V_j$ and ending inside $W_j$ intersects a compact of ${\mathcal {I}}_{j}$;

• (c) starting inside $W_j$ and ending outside $V_j$ intersects a compact of ${\mathcal {O}}_j$;

and for any $j\in \{1,\ldots ,r\}$ and for any compact sets $I\subset {\mathcal {I}}_{j}$ and $O\subset {\mathcal {O}}_{j}$, there exists a pseudo-orbit with jumps inside the tiles of ${\mathcal {C}}$, with starting point in $I$ and ending point in $O$. Roughly speaking, given a covering family of ${\mathcal {E}}_{H,e}$ outside ${\mathcal {V}}$, any pseudo-orbit either returns regularly to the tiled cross sections, during the time it passes out of ${\mathcal {V}}$ or else, intersects a compact set $F_j\subset V_j$. In this last situation, the pseudo-orbit must go through an “in set” $I\subset {\mathcal {I}}_j$ and then through an “out set” $O\subset {\mathcal {O}}_j$. Moreover, we can even switch the segment of the pseudo-orbit between $I$ and $O$ by a pseudo-orbit with jumps inside the tiles of ${\mathcal {C}}$. The existence of these objects follows, up to considering cross sections, from  [5], Sect. 4] for symplectomorphisms.

1.3 Connecting Pseudo-Orbits

Arnaud et al. proved, in [5, Proposition 4.2], that if the eigenvalues of any closed orbit of a symplectomorphism do not satisfy non-trivial resonance relations, then the closed orbits are uniformly avoidable. Therefore, since the transversal linear Poincaré flow is a symplectomorphism, then for any $H\in C^2(M,{\mathbb {R}})$ and any periodic point $p$ of $H$ with period $\ell$, the eigenvalues of $\Phi _H^{\ell }(p)$ do not satisfy non-trivial resonances, then the closed orbits of $H$ are uniformly avoidable.

Hence, to prove the Connecting Lemma for pseudo-orbits of Hamiltonians it is enough to show that if $Per(H)$ on ${\mathcal {E}}_{H,e}$ are uniformly avoidable, then, for any $C^2$-neighborhood ${\mathcal {U}}$ of $H$ and for any $x,y\in {\mathcal {E}}_{H,e}$, there is $\tilde{H}\in {\mathcal {U}}$ and $t>0$, such that $\tilde{H}(x)=e$ and $X^t_{\tilde{H}}(x)=y$. It is obvious that this statement follows immediately if $y\in {\fancyscript{O}}_H(x)$. In fact, to prove the Connecting Lemma, it is enough to show it for certain points $x,y\in {\mathcal {E}}_{H,e}$. Indeed, the same argument as the ones in [5, Lemma 3.12] allows us to reduce the proof to the case when $x,y$ are not closed orbits. So, we assume that $x,y\notin Per(H)$ and $y\notin {\fancyscript{O}}_H(x)$ for every $H$ in a $C^2$-neighborhood ${\mathcal {U}}_0$. Using Kakutani towers for flows ([4, 6]) and adapting the ideas of the proof in [5, Proposition 3.13] we obtain that there exist a neighborhood ${\mathcal {U}}\subset {\mathcal {U}}_0$ of $H$, a family of disjoint open sets ${\mathcal {V}}$ and a family of perturbation flowboxes $({\mathcal {F}}_H({\mathcal {C}}_i,T))_i$ for $(H,{\mathcal {U}})$ with disjoint supports, both ${\mathcal {V}}$ and $({\mathcal {F}}_H({\mathcal {C}}_i,T))_i$ not containing $x$ nor $y$, such that the perturbation flowboxes covering ${\mathcal {E}}_{H,e}$ outside ${\mathcal {V}}$. Now, the classical strategy developed in  [10] and carried out in the proof of [5, Proposition 3.4] for symplectomorphisms allows us to obtain a pseudo-orbit preserving the tiling and, consequently one erases, flowbox after flowbox, all the jumps of the pseudo-orbit. Thus, there exist $\tilde{H}\in {\mathcal {U}}$ and $t>0$, such that $\tilde{H}(x)=e$ and $X^t_{\tilde{H}}(x)=y$.