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In this paper, it is shown that for a minimal system ( X ,  G ), if  H  is a normal subgroup of  G  with finite index  n , then  X  can be decomposed into  n  components of closed sets such that each component is minimal under  H -action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets