Multivariate Frequent Stability and Diam-Mean Equicontinuity

Abstract In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity — called “frequent m -stability” and “diam-mean m -equicontinuity”, where $$m &gt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is a natural number. Here, “multivariate” refers to the dynamical behaviour of tuples of starting points. Systems for which the factor map to the maximal equicontinuous factor () is finite-to-one for a residual set — “almost finite-to-one extensions” — or a set of full measure — “almost-surely finite-to-one extensions” — are respectively characterised by these dynamical rigidity properties. In the case of a second-countable, locally compact, abelian acting group and a natural number m , a minimal system is frequently $$(m+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -stable if and only if it is an almost m -to-1 extension of its . Similarly, it is shown that a minimal system is diam-mean $$(m+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -equicontinuous if and only if it is an almost-surely m-to-1 extension of its .