Length functions, multiplicities and algebraic entropy

Abstract

We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of -modules. Then entropy will be viewed as a function on modules over the polynomial ring 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 -modules. Furthermore, if one requires that the induced length function 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 , generalizing a result proved for Abelian groups by A. Giordano Bruno.