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 \(m > 1\) 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 (mef) 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 \((m+1)\) -stable if and only if it is an almost m-to-1 extension of its mef. Similarly, it is shown that a minimal system is diam-mean \((m+1)\) -equicontinuous if and only if it is an almost-surely m-to-1 extension of its mef.
