Topological weak mixing and diffusion at all times for a class of Hamiltonian systems

We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.


Introduction
KAM theory (after Kolmogrov, Arnol'd, and Moser) states that, under mild non-degeneracy assumptions, Hamiltonian systems close to integrable have their phase space almost completely filled by invariant quasi-periodic tori. Starting from three degrees of freedom for autonomous Hamiltonians the existence of such tori on an energy surface does not prevent the orbits from circulating between the tori inside the surface. Indeed, it was conjectured by Arnol'd that a "general" Hamiltonian should have a dense orbit on a "general" energy surface [A]. A great amount of work has been dedicated to proving this conjecture (giving a precise meaning to the word "general"), but the picture is not yet completely clear, especially when it comes to real analytic Hamiltonians (see for example [BKZ] and references therein).
In his ICM list of problems [H], M. Herman asks: Can one find an example of a C ∞ -Hamiltonian H in a small C k -neighborhood, k ≥ 2, of H 0 = r 2 /2 such that on the energy surface {H = 1} the Hamiltonian flow has a dense orbit?
A remarkable result in this direction is due to [KZZ]: they present an example of a Hamiltonian H of the form H(θ, r) = r,r 2 + h(r, θ) ∈ C ∞ with a trajectory dense in a subset of the energy surface of large measure.
Here we present examples, with a degenerate integrable part, but with even more chaotic behavior for the perturbed system. Namely, we show that in a class of perturbations of rotators, the generic Hamiltonian is topologically weakly mixing on every energy surface. We also give examples of perturbed rotators for which the dynamics is diffusive at all times.
We now give the exact definitions of these properties and state our results precisely.
Definition 1 (Topological weak mixing). We say that the flow Φ t H is topologically weakly mixing if there exists a sequence (t n ) n∈N such that for any two open sets A, B on the same (arbitrarily chosen) energy surface E H (c) = {(θ, r) | H(θ, r) = c}, there exists N = N (A, B) such that Φ tn H (A) ∩ B = ∅ for all n ≥ N. Definition 2 (Diffusion at all times). We say that the flow Φ t H exhibits diffusion at all times if for any open set A ⊂ T d × R d and any R ≥ 0 there exists T = T (A, R) such that |π r (Φ t H (A))| > R for all t ≥ T . Here π r stands for the projection onto the r-variables. Given ρ > 0, denote by C ω ρ the space of bounded real functions on R d × R d , that are Z dperiodic in the first d-vector of components, and can be extended to holomorphic functions δ} be a neighborhood of zero in the space of analytic functions with the above norm. This is a Baire space. Fixing ρ = 1 without loss of generality, we will write (Y continuum) such that for any ω ∈ Y , for any δ > 0, there exists a real analytic function h : T d → R, such that h ∈ O ω δ , with the following property: the Hamiltonian flow Φ t H (θ, r) defined by the Hamiltonian exhibits diffusion at all times.
By a generic set in this paper we mean a dense G δ set.
Note that the energy surface in the statement above is unbounded, which drove us to treat the topological version of weak mixing rather than the classical notion.
Diffusion at all times shows that the rigidity property of the rotator (convergence of the dynamics to identity along a subsequence of times) can be destroyed on all energy surfaces by a small perturbation. However, our examples are not topologically mixing, and it would be very interesting to produce examples that are topologically mixing on energy surfaces.
An interesting question concerns the possibility of similar examples in a neighborhood of an elliptic equilibrium. If the components of the frequency vector at the equilibrium are all of the same sign, the energy surfaces are bounded. We hope that the methods of [FS] can be used to construct smooth examples of topologically weakly mixing Hamiltonians in this context as well.
From the form of the perturbations of the rotator in Theorems A and B, we can see that the Hamiltonians that we construct are in fact in the closure of Hamiltonians that are conjugate to the rotator. Just like reparametrizations of linear flows on the torus, or abelian skew products above them, our constructions can thus be viewed a posteriori as particular instances of the AbC method (Approximation by Conjugation) [AK], that is also called AK-method in reference to Dmitry Anosov and Anatoly Katok who first introduced the method. The method was already used in the Hamiltonian context by Katok in [K2] to show the existence of integrable degenerate Hamiltonians with some particular Liouville frequencies and bounded energy surfaces that can be smoothly perturbed to become ergodic on the energy surfaces. Subsequent constructions that use the AbC method with several frequencies appeared in [EFK, FS, FF] to discuss the stability of elliptic equilibria and invariant tori, in particular those with Diophantine frequency vectors. The examples of [K2] were constructed following the usual AK-method with successive conjugations of a circle action, which gives C ∞ flows that are rigid in the sense that the dynamics converge to Identity along a subsequence of times. In our constructions, we bypass the smoothness limitation and the rigidity of the perturbed dynamics by resorting to the reparametrization technique of translation flows in dimension larger than 3 used in [F]. This technique exploits the Liouville phenomenon in several directions [Y] to avoid the Denjoy-Koksma cancellations that appear in dimension two [K1,Koc].

Notations and definitions
To alleviate the notations, we will give the proofs for d = 3 since there is no difference at all in the proof of the general case.

General notations
-For a vector r, its components are denoted by r j , j = 1, 2, 3; for a vector r 0 ∈ R 3 , we denote its components by r 0,j , j = 1, 2, 3.
-For a set S in the phase space, let π r (S) stand for the orthogonal projection of S onto the space of actions (r-space). In the similar way introduce notations π θ (S), π r j (S), π θ j (S) for j = 1, . . . , d.
-When we write that p q is a rational number or p q ∈ Q, we assume that q ∈ N, q ≥ 1, p ∈ Z, and the numbers p and q are relatively prime.

Arithmetic reminders. Yoccoz pairs of frequencies
For an irrational number α there exists a sequence of rational numbers ( pn qn ) n≥1 , called the convergents of α such that |||q n−1 α||| < |||kα||| for all k < q n , and for any n 1 where α and α ′ are irrational real numbers with the corresponding sequences of convergents ( pn qn ) n≥1 and ( p ′ n q ′ n ) n≥1 . We say that ω = (α, α ′ , 1) ∈ Y if for all n = 0, . . . , ∞, the denominators of the convergents of α and α ′ , respectively, satisfy: By [Y], the set Y is nonempty, of cardinality continuum. This is the set of frequencies used in Theorem A.

Intervals and rectangles in an energy surface
Here we describe the standard sets used in the construction. In particular, intervals are defined to be small one dimensional curves that lie in a given energy surface and whose projection onto the 5-dimensional space (θ, r 1 , r 2 ) is a linear segment parallel either to the θ 1 axis or to the θ 2 axis. The coordinate r 3 is defined by the requirement that the curve lies in the energy surface. More precisely: -Given s 0 = (θ 0 , r 0 ) and l > 0, define the intervals -For any s 0 = (θ 0 , r 0 ), l 1 > 0 and l 2 > 0, define the rectangle As before, the projection of R(θ 0 , r 0 , l 1 , l 2 ) onto the space (θ, r 1 , r 2 ) is a flat rectangle parallel to (θ 1 , θ 2 )-plane; r 3 is chosen so that R(θ 0 , r 0 , l 1 , l 2 ) ⊂ E H (c).
We say that the size of the rectangle R(θ 0 , r 0 , l 1 , l 2 ) is l 1 × l 2 .
-Given n and These sets, having full dimension in E H (c), will be used as test sets: in particular, to prove Theorem B, we will show that at certain times t n , the image of any rectangle To do so, we will need the notion of stretching: Definition 4. Given positive l, L and t, we say that the flow map Φ t H is (1, l, L)-stretching if for any interval J (1) = J (1) (θ 0 , r 0 , l) with |r 0 | ≤ L/10 we have: and the map (θ, r) → π r 2 (Φ t H (θ, r)) is independent of θ 1 .

Proofs of the main theorems
Here we prove the main theorems modulo the technical statements, whose demonstration is deferred to the next section.

The construction for Theorem A
Let us fix an arbitrary vector (α,

Theorem A follows from
Theorem 1. For any ω ∈ Y , for h as in (2.1), the Hamiltonian flow Φ t H (θ, r) defined by the Hamiltonian exhibits diffusion at all times.
Remark 1. This Hamiltonian can be seen as a limit of an Anosov-Katok type construction, i.e., it has the form: where Ψ i are symplectic analytic coordinate changes.
The proof of Theorem 1 relies on the following proposition that is proved in Section 3.1.
satisfies for each n: Here we show how this proposition implies Theorem 1.
Proof of Theorem 1. Since the sequences (q n ) and (q ′ n ) satisfy (1.2), the union of the intervals ∪ n>N [e qn , q n+1 /4] ∪ [e q ′ n , q ′ n+1 /4] contains the half-line t > e q N . Proposition 1 implies that for each t ∈ [e qn , q n+1 /4], Φ t H stretches small rectangles in the direction of r 1 with a large factor, and for each t ∈ [e q ′ n , q ′ n+1 /4], Φ t H stretches small rectangles in the direction of r 2 .
Hence Φ t H exhibits stretching with an increasingly strong factor as t → ∞, in at least one of the two directions r 1 and r 2 . This implies the conclusion of Theorem 1. ✷
Lemma 1. There exists a generic (dense G δ ) setŜ ⊂ S ⊂ R 2 of pairs (α, α ′ ) satisfying the following: there exist sequences (p n /q n ) n≥1 and (p ′ n /q ′ n ) n≥1 of rational numbers such that estimate (2.2) holds for all n.
Proof. We want to describe the set S of pairs (α, α ′ ) such that for any N there exist p/q and p ′ /q ′ ∈ Q such that q > N, q ′ ≥ q 4 , p, p ′ ∈ Z, and The set S contains the following setŜ: which is a countable intersection (in N) of open dense sets.
Here we present an explicit example of a Hamiltonian whose flow is topologically weakly mixing. It is easy to see that such examples can be produced arbitrarily close to H 0 . From this, the genericity of Hamiltonians with the weak mixing property is obtained in the standard way.

Proof of Theorem B.
It follows from classical arguments (Cf. [Ha]) that weak mixing for the flows as in Theorem B holds for a G δ -set of functions (h 1 , h 2 ) ∈ O ω δ (0) 2 . It is left to show the density of weak mixing for (h 1 , h 2 ) ∈ O ω δ (0) 2 for a fixed δ. To do this, notice that for any ǫ > 0, both (φ(θ) − 1) andh(θ) can be chosen ǫ-close to zero in the fixed norm: it is enough to choose q 1 large enough. Moreover, from the proof of Theorem 2 it follows that the same result holds true if we change φ(θ) andh(θ) by φ(θ) + P (θ) ∈ O ω δ (0) andh(θ) + Q(θ) ∈ O ω δ (0) with P and Q trigonometric polynomials. This implies the density of the weak mixing property. ✷ The proof of Theorem 2 relies on the following two propositions that are proved in Sections 3.1 and 3.2, respectively.
Proposition 2. For ω as in (2.2), φ as in (2.3),h as in (2.4),H as in (2.5), and t n = e q ′ n , we have: Proposition 3. For ω as in (2.2), φ as in (2.3),h as in (2.4), the Hamiltonian flow Φ tH (θ, r) defined byH satisfies for t n = e q ′ n the following. For any rectangle R n := R(θ 0 , r 0 , 1/q n , 1/q n ) with |r 0 | ≤ n and any box B n (see notations in Sec. 1.3) there exists a rectangle R ′ n ⊂ R n of size 1/q 3 n × 1/q 3 n such that Proof of Theorem 2. Fix R n and B n as above.
By Proposition 3, there exists a rectangle R ′ n ⊂ R n of size 1/q 3 n × 1/q 3 n such that By Proposition 2, we can find a rectangleR n ⊂ R ′ n such that π r (Φ tñ H (R n )) ⊂ π r (B n ).
Hence Φ tñ H (R n )) ⊂ B n and the proof is finished. ✷ 3 Stretching 3.1 Stretching in the action directions.
Since |c 1 | < q/10, we get π r 1 (Φ t H (s + )) > 2q. In the same way, there is a point s − ∈ J (1) such that π r 1 (Φ t H (s − )) < −2q. The result follows by continuity. ✷ Here we prove Proposition 1 that was used for the proof of Theorem 1.
Proof of Proposition 1.
By the same argument, r 1 (s − , t) ≤ −q n . Thus, Φ t H is (1, 1/q n , q n ) stretching. ✷ Below we prove an analog of Proposition 1 for the HamiltonianH of Theorem 2. To begin with, notice that our choice of φ andh implies that the Hamiltonian system ofH has a particularly simple form.
Lemma 3. For ω as in (2.2), φ as in (2.3),h as in (2.4), the Hamiltonian flow Φ tH (θ, r) defined byH where C =H(θ, r). We omit the expression for r 3 since it is not used below.
Proof of Lemma 3. Recall that the value ofH(θ, r) = 1 φ(θ) ( r, ω +h(θ)) := C is constant on the solutions of the corresponding system of equations. For j = 1, 2 we have: Explicit substitution finishes the proof. ✷ In the next lemma (which is an analog of Lemma 2) we study the action components of the above system in a simplified form: we consider only the n-th term in the sums above.
The study of the angle components is postponed to Proposition 3.

✷
Proof of Proposition 2. The proof of Proposition 2 follows from Lemma 4 exactly as Proposition 1 followed from Lemma 2. ✷

Stretching in the angle directions
In this section we prove Proposition 3. Namely, we study the behavior of π θ (Φ tH ), which is the solution of the equationθ = 1 φ(θ) ω.
( 3.4) Since this flow does not depend on r, we fix an arbitrary r 0 and omit it from the notations.
Proof of Proposition 3. Sinceh(θ) does not depend on the r variables, the restriction of the flow ofH = 1 φ(θ) r, ω +h(θ) onto T 3 is the same as that ofH = 1 φ(θ) r, ω . We will study the latter one.