‘Growths of endomorphisms of finitely generated semigroups’
[with V. Maltcev]
Journal of the Australian Mathematical Society, 102, no. 2 (April 2017), pp. 163–184.
DOI: 10.1017/S1446788716000264.


This paper studies the growth of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of a semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms on homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytic tricks. We state various problems and suggestions for future research.