The essential coexistence phenomenon in dynamics

Abstract

We construct an example of a Hamiltonian flow f t on a 4-dimensional smooth manifold M which after being restricted to an energy surface M e demonstrates essential coexistence of regular and chaotic dynamics that is there is an open and dense f tinvariant subset U ⊂ M e such that the restriction f t |U has nonzero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary ∂U , which has positive volume all Lyapunov exponents of the system are zero.

A volume preserving flow with essential coexistence of zero and non-zero Lyapunov exponents

“…(2) g t = f t outside N × U 0 , and hence g t is a gentle perturbation of f t and satisfies Statements (3)- (5) of Proposition 3.1; (3) g t preserves the subbundles E ω f , ω = ua, uab, uabτ ; moreover, (4.7)…”

“…It remains to prove Statements (4) and (5). We need the following lemma showing that Π 0 is a global cross-section for the flow g t |N × U 0 .…”

“…While this paper deals only with dynamical systems with continuous time, it is worth mentioned that in the discrete-time case essential coexistence of chaotic and regular behavior have been demonstrated in various situations by Przytycki and Liverani for area preserving diffeomorphisms and by Bunimovich for billiards; see the paper [5], which surveys recent result on essential coexistence, and references therein.…”

“…According to Lemma B.5, M 0 depends only on the Riemannian metric and the start-up flow f t .Let the number λ > 0 and the subset Π ⊂ Π 0 be as in Statement(5) of Proposition 4.2. Given K > 0, let (4.16) Λ = Λ (K) = {z ∈ Π :1 k log d g k (z, v) − λ ≤ 0.1λ, for all v ∈ E ua f (z), v = 1 and all |k| ≥ 0−i Λ (K),where k 0 > 0 is a number which will be defined later (seeLemma 4.5).…”

Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

“…Later, on magnifying the picture between KAM curves and especially near the hyperbolic periodic points H k , we found chaotic regions. We believe that the map T gives a very important geometric example of coexistence of regular and chaotic behaviors (we refer to an important survey papers [5] [11] on the coexistence problem).…”