Product recurrence and distal points

Abstract

Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Čech compactification of an acting group.) If the semigroup <italic>E</italic> acts on the space <italic>X</italic> and <italic>F</italic> is a closed subsemigroup of <italic>E</italic>, then <italic>x</italic> in <italic>X</italic> is said to be <italic>F</italic>-recurrent if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p x equals x"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">px = x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of upper F"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p \in F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and product <italic>F</italic>-recurrent if whenever <italic>y</italic> is an <italic>F</italic>-recurrent point (in some space <italic>Y</italic> on which <italic>E</italic> acts) the point (<italic>x, y</italic>) in the product system is <italic>F</italic>-recurrent. The main result is that, under certain conditions, a point is product <italic>F</italic>-recurrent if and only if it is a distal point.