We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of ROEX -modules. Then entropy will be viewed as a function h L on modules over the polynomial ring ROEX extending L. In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R-modules to length functions on ROEX -modules. Furthermore, if one requires that the induced length function h L satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L-entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value 1, generalizing a result proved for Abelian groups by A. Giordano Bruno.

The new notion of intrinsic algebraic entropy (ent) over tilde of endomorphisms of Abelian groups is introduced and investigated. The intrinsic algebraic entropy is compared with the algebraic entropy, a well-known numerical invariant introduced in the sixties and recently deeply studied also in its relations to other fields of Mathematics. In particular, it is shown that the intrinsic algebraic entropy and the algebraic entropy coincide on endomorphisms of torsion Abelian groups, and their precise relation is clarified in the torsion-free case. The Addition Theorem and the Uniqueness Theorem are also proved for ent, in analogy with similar results for the algebraic entropy. Furthermore, a relevant connection of eat to the algebraic entropy of a continuous endomorphism of a locally compact Abelian group G is pointed out; this allows for the computation of the algebraic entropy in case G is totally disconnected

In 1965 Adler, Konheim and McAndrew defined the topological entropy of a continuous self-map of a compact space. In 1971 Bowen extended this notion to uniformly continuous self-maps of (not necessarily compact) metric spaces and this approach was pushed further to uniform spaces and topological groups by many authors, giving rise to various versions of the topological entropy function. In 1981 Peters proposed a completely different (algebraic) entropy function for continuous automorphisms of non-compact LCA groups. The aim of this paper is to discuss some of these notions and their properties, trying to describe the relations among the various entropies and to correct some errors appearing in the literature

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Let R be a ring, let G be an amenable group and let R˚G be a crossed product. The goal of this paper is to construct, starting with a suitable additive function L on the category of left modules over R, an additive function on a subcategory of the category of left modules over R˚G, which coincides with the whole category if LpRRq ă 8. This construction can be performed using a dynamical invariant associated with the original function L, called algebraic L-entropy. We apply our results to two classical problems on group rings: the Stable Finiteness and the Zero-Divisors Conjectures.-----------