Vanishing of the entropy pseudonorm for certain integrable systems

Abstract

We introduce the notion of entropy pseudonorm for an action of R and prove that it vanishes for the group actions associated with a big class of integrable Hamiltonian systems. 1. Entropy pseudonorm Let W be a smooth manifold and Φ : (R, +) → Diff(W ) a smooth action on it. Assume there exists a compact Φ-invariant exhaustion of W . Define the following function on R (where htop is the topological entropy): ρΦ(v) = htop(Φ(v)), v ∈ R. This function is a pseudonorm on R (ρΦ(v) is well-defined because with our hypothesis the entropy hd of [Bo] does not depend on the distance function d, homogeneity is standard and the triangle inequality follows from the Hu formula [H]). We call ρΦ the entropy pseudonorm. We will investigate it in the case of the Poisson action corresponding to an integrable Hamiltonian system on a symplectic manifold (W , ω). Namely, let (W , ω) possess pair-wise Poisson commuting functions I1, I2, . . . , In, which are functionally independent almost everywhere. Denote by φi the time τ shift along the Hamiltonian vector field of the function Ii. The maps φi commute and therefore generate the Poisson action of the group (R, +), Φ(τ1, . . . , τn) def = φ1 1 ◦ · · · ◦ φn n : W 2n → W , with the corresponding momentum map Ψ = (I1, . . . , In) : W 2n → R, see [A]. The entropy pseudonorm ρΦ vanishes in the following important cases: − Williamson-Vey-Eliasson-Ito non-degenerate singularities [E, I]; − Taimanov non-degeneracy condition [T]. In the first case vanishing of topological entropy of the Hamiltonian flow was proved in [P2], in the second case in [T]. Since there is nothing special about the Hamiltonian in these situations, it can be changed to any of the integrals and ρΦ ≡ 0 follows. Also in [P1, BP]) vanishing of htop was proven for the cases: − Systems integrable with periodic integrals; − Collectively integrable systems (the definition is in [GS]). It is not difficult to see that in both cases the entropy pseudonorm ρΦ vanishes as well. Note that Liouville integrability does not imply vanishing of topological entropy, see [BT] (more examples in [Bu]). For these examples the entropy pseudonorm is degenerate, but it is possible to construct integrable examples [K] such that ρΦ is a norm. In the present paper we prove vanishing of the entropy pseudonorm for another class of integrable systems. These systems were recently actively studied in mathematical physics in the