English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

Vanishing of the entropy pseudonorm for certain integrable systems