The addition theorem for locally monotileable monoid actions

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem

“…As an application of the Addition Theorem for h top and of the Bridge Theorem, we show that also h alg satisfies an Addition Theorem for left S -actions S λ X on discrete Abelian groups, so far known only under the hypotheses that either X is torsion [10] or S is locally monotileable [11]: Algebraic Addition Theorem. Let S be a cancellative and right amenable monoid, X an Abelian group, S λ X a left S -action and Y an S -invariant subgroup of X. Then,…”

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For a left action S λ X of a cancellative right amenable monoid S on a discrete Abelian group X, we construct its Ore localization G λ * X * , where G is the group of left fractions of S ; analogously, for a right action K ρ S on a compact space K, we construct its Ore colocalization K * ρ * G. Both constructions preserve entropy, i.e., for the algebraic entropy h alg and for the topological entropy h top one has h alg (λ) = h alg (λ * ) and h top (ρ) = h top (ρ * ), respectively.Exploiting these constructions and the theory of quasi-tilings, we extend the Addition Theorem for h top , known for right actions of countable amenable groups on compact metrizable groups [36], to right actions K ρ S of cancellative right amenable monoids S (with no restrictions on the cardinality) on arbitrary compact groups K.When the compact group K is Abelian, we prove that h top (ρ) coincides with h alg (ρ ∧ ), where S ρ ∧ X is the dual left action on the discrete Pontryagin dual X = K ∧ , that is, a so-called Bridge Theorem. From the Addition Theorem for h top and the Bridge Theorem, we obtain an Addition Theorem for h alg for left actions S λ X on discrete Abelian groups, so far known only under the hypotheses that either X is torsion [10] or S is locally monotileable [11].The proofs substantially use the unified approach towards entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids, as developed in [17] (for N-actions) and [58].

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“…When the compact group K is Abelian, we prove that h top (ρ) coincides with h alg (ρ ∧ ), where S ρ ∧ X is the dual left action on the discrete Pontryagin dual X = K ∧ , that is, a so-called Bridge Theorem. From the Addition Theorem for h top and the Bridge Theorem, we obtain an Addition Theorem for h alg for left actions S λ X on discrete Abelian groups, so far known only under the hypotheses that either X is torsion [10] or S is locally monotileable [11].…”