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Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

100%

Downarowicz T. , Kasjan S.

2015

tom 229

nr 1

45-72

Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do all generalized Sturmian subshifts (not only the classical Sturmian subshift). We also give an example of an irregular Toeplitz subshift which meets our criterion. We give an example of a model of an odometer which is not even Toeplitz (it is weakly mixing), hence does not meet our criterion. However, for this example, we manage to produce a separate proof of Sarnak's conjecture. Next, we provide a class of Toeplitz sequences which fail Sarnak's conjecture (in a weak sense); all these examples have positive entropy. Finally, we examine the example of a Toeplitz sequence from [AKL] (which fails Sarnak's conjecture in the strong sense) and prove that it also has positive entropy (this proof has been announced in [AKL]). This paper can be considered a sequel to [AKL], it also fills some gaps of [D].

On two tame algebras with super-decomposable pure-injective modules

100%

Kasjan S. , Pastuszak G.

Colloquium Mathematicum

2011

tom 123

nr 2

249-276

Let k be a field of characteristic different from 2. We consider two important tame non-polynomial growth algebras: the incidence k-algebra of the garland 𝒢₃ of length 3 and the incidence k-algebra of the enlargement of the Nazarova-Zavadskij poset 𝒩 𝓩 by a greatest element. We show that if Λ is one of these algebras, then there exists a special family of pointed Λ-modules, called an independent pair of dense chains of pointed modules. Hence, by a result of Ziegler, Λ admits a super-decomposable pure-injective module if k is a countable field.

On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth

100%

Kasjan S. , Pastuszak G.

tom 136

nr 2

179-220

Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pure-injective module if k is a countable field.