Abstract. For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σ λ of the direct sum Γ K is the endomorphism defined by (x i ) i∈Γ → (x λ(i) ) i∈Γ [3]. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function λ. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.We denote by Z, P, and N respectively the set of integers, the set of primes, and the set of natural numbers; moreover N 0 = N ∪ {0}. For a set Γ, P fin (Γ) denotes the family of all finite subsets of Γ. For a set Λ and an abelian group G we denote by G Λ the direct product i∈Λ G i , and by G (Λ) the direct sum i∈Λ G i , where all G i = G. For a set X, n ∈ N, and a function f : X → X let Per(f ) be the set of all periodic points and Per n (f ) the set of all periodic points of order at most n of f in X.

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

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