This paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups:
a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szép product of an
arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary,
not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some
new results concerning the strong actions of P2 on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these
repesentations and show that the structure of the representation can be explained by studying the binary representations of
Authors:Shigeki Akiyama, Tibor Borbély, Horst Brunotte, Attila Pethő, and Jörg M. Thuswaldner
Summary We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.
We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse
semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of
Kawamura on certain kinds of representations of the Cuntz C*-algebras and to the branching function systems of Bratteli and Jorgensen.
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary
methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with
respect to θ for k = 2.