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

Abstract

We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

The essential coexistence phenomenon in dynamics

“…In [CHP13b], the first three authors of this paper constructed a volume preserving C ∞ flow f t on a 5-dimensional manifold with essential coexistence. Moreover, in that paper, U c is a union of 3-dimensional invariant submanifolds and f t is a linear flow with a Diophantine frequency vector on each invariant submanifold.…”

“…First examples of systems with discrete and continuous time demonstrating essential coexistence, which are volume preserving, were constructed in [HPT13,CHP13b,Che12] (see also [CHP13a] for a survey of recent results). Naturally, one would like to construct examples of systems with essential coexistence which are Hamiltonian.…”

“…In the course of our construction we also obtain an area preserving C ∞ diffeomorphism of a 2-dimensional torus as well as a volume preserving C ∞ flow on a 3-dimensional manifold both demonstrating the essential coexistence phenomenon, see Theorems C and B respectively. These examples are simpler than the ones in [HPT13] and [CHP13b], where the corresponding constructions require 5-dimensional manifolds.…”

SRB measures for pointwise hyperbolic systems on open regions

“…The notion of gentle perturbations is used in [9,15]. The definition we give here is slightly different.…”

“…Furthermore, by using Katok's map, it is proved that every smooth compact manifold of dimension greater than two carries a hyperbolic diffeomorphism (see [4,5,11], and see also [15] for a flow version). Chen et al [9,10] and Hu et al [16] constructed systems that exhibit essential coexistence of chaotic and nonchaotic behaviors, where the chaotic parts are pointwise partially hyperbolic diffeomorphisms or pointwise hyperbolic flows. In the above examples, the systems are volume preserving.…”