The notion of algebraic entropy has a long history dating back to a very brief outline of the basic ideas in a paper by R. L. Adler, A. G. Konheim and M. H. McAndrew in 1965. The notion was subsequently developed by a long series of authors working in various contexts. For an abelian group A and an endomorphism φ: A → A, the algebraic entropy halg (φ) is defined by 1 halg (φ) = supF∈F(A) limn→∞ nlog|F+φ(F) +. . .+φn−1 (F )|, where F(A) is the family of all finite non-empty subsets of A. The algebraic entropy of an m-tuple of commuting endomorphisms of an abelian group is defined similarly. Under the celebrated Kaplansky paradigm, abelian groups provided with an m-tuple of pairwise commuting endomorphisms can be identified with Z[X1, . . ., Xm ]-modules, so the algebraic entropy becomes a length function h of Z[X1, . . ., Xm ]-modules. We show that this length function h can be completely described via the Mahler measure of polynomials f ∈ Z[X1, . . ., Xm ]. As an application we obtain various uniqueness theorems for h. To ease the readers non-familiar with the dynamical aspects of the paper, we first provide a treatment of the entropy of only single endomorphisms, leaving the general case of entropy of a finite set of pairwise commuting endomorphisms for the final part.
A length function of Z[X1, . . ., Xm ]-modules and Mahler measure
Bruno A. G.;Spirito D.;
2026-01-01
Abstract
The notion of algebraic entropy has a long history dating back to a very brief outline of the basic ideas in a paper by R. L. Adler, A. G. Konheim and M. H. McAndrew in 1965. The notion was subsequently developed by a long series of authors working in various contexts. For an abelian group A and an endomorphism φ: A → A, the algebraic entropy halg (φ) is defined by 1 halg (φ) = supF∈F(A) limn→∞ nlog|F+φ(F) +. . .+φn−1 (F )|, where F(A) is the family of all finite non-empty subsets of A. The algebraic entropy of an m-tuple of commuting endomorphisms of an abelian group is defined similarly. Under the celebrated Kaplansky paradigm, abelian groups provided with an m-tuple of pairwise commuting endomorphisms can be identified with Z[X1, . . ., Xm ]-modules, so the algebraic entropy becomes a length function h of Z[X1, . . ., Xm ]-modules. We show that this length function h can be completely described via the Mahler measure of polynomials f ∈ Z[X1, . . ., Xm ]. As an application we obtain various uniqueness theorems for h. To ease the readers non-familiar with the dynamical aspects of the paper, we first provide a treatment of the entropy of only single endomorphisms, leaving the general case of entropy of a finite set of pairwise commuting endomorphisms for the final part.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
