A generic incompressible flow is topological mixing

Abstract

In this Note we prove that there exists a residual subset of the set of divergence-free vector fields defined on a compact, connected Riemannian manifold M , such that any vector field in this residual satisfies the following property: Given any two nonempty open subsets U and V of M , there exists <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>τ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∩</mml:mo> <mml:mi>V</mml:mi> <mml:mo>≠</mml:mo> <mml:mo>∅</mml:mo> </mml:math> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>t</mml:mi> <mml:mo>⩾</mml:mo> <mml:mi>τ</mml:mi> </mml:math> .