Ubiquity of entropies of intermediate factors

We consider topological dynamical systems ( X , T ) where is a compact metrizable space and denotes an action of countable amenable group G on by homeomorphisms. For two such Y S factor map π : → intermediate system Z R for which can be written as composition maps ψ φ . In this paper, we show that any -subshifts the set entropies subshift factors dense in interval [ h ] As corollary, also prove if are zero-dimensional -systems, then equal to Our proofs rely generalized Marker Lemma may independent interest.