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1
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Reading along arithmetic progressions

100%
Downarowicz T.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 2
293-296
EN
Given a 0-1 sequence x in which both letters occur with density 1/2, do there exist arbitrarily long arithmetic progressions along which x reads 010101...? We answer the above negatively by showing that a certain regular triadic Toeplitz sequence does not have this property. On the other hand, we prove that if x is a generalized binary Morse sequence then each block can be read in x along some arithmetic progression.
2
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A non-regular Toeplitz flow with preset pure point spectrum

64%
Downarowicz T., Lacroix Y.
Studia Mathematica
|
1996
|
tom 120
|
nr 3
235-246
EN
Given an arbitrary countable subgroup $σ_0$ of the torus, containing infinitely many rationals, we construct a strictly ergodic 0-1 Toeplitz flow with pure point spectrum equal to $σ_0$. For a large class of Toeplitz flows certain eigenvalues are induced by eigenvalues of the flow Y which can be seen along the aperiodic parts.
3
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Weak Closure Theorem fails for ℤ²-actions

64%
Downarowicz T., Kwiatkowski J.
Studia Mathematica
|
2002
|
tom 153
|
nr 2
115-125
EN
We construct an example of a Morse ℤ²-action which has rank one and whose centralizer contains elements which cannot be weakly approximated by the transformations of the action.
4
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Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows

64%
Downarowicz T., Lacroix
Studia Mathematica
|
1998
|
tom 130
|
nr 2
149-170
EN
Let $(Z,T_Z)$ be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of $(Z,T_Z)$ is Borel isomorphic to an almost 1-1 extension of $(Z,T_Z)$. Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz flow.
5
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A criterion for Toeplitz flows to be topologically isomorphic and applications

51%
Downarowicz T., Kwiatkowski J., Lacroix Y.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
219-228
EN
A dynamical system is said to be coalescent if its only endomorphisms are automorphisms. The question whether there exist coalescent ergodic dynamical systems with positive entropy has not been solved so far and it seems to be difficult. The analogous problem in topological dynamics has been solved by Walters ([W]). His example, however, is not minimal. In [B-K2], a class of strictly ergodic (hence minimal) Toeplitz flows is presented, which have positive entropy and trivial topological centralizers (the last condition implies coalescence). The entropy, however, is only estimated from below. Also the class is obtained in a not completely constructive way. The basic idea of this paper is contained in Section 2, in a criterion which describes homomorphisms (isomorphisms) between Toeplitz flows in terms of a block code simplified to a function sending blocks of a given length to blocks of the same length. This idea is then applied in Section 3 to effectively construct an uncountable family of pairwise nonisomorphic Toeplitz flows with topological entropy equal to a common arbitrarily preset value. Furthermore, all the Toeplitz flows have the same maximal uniformly continuous factor. In Section 4 we obtain conditions sufficient for coalescence of a Toeplitz flow. In particular, all Toeplitz flows of Section 3 turn out to be coalescent. We are grateful to Professor P. Liardet for several helpful conversations and valuable remarks.
6
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Spectral isomorphisms of Morse flows

51%
Downarowicz T., Kwiatkowski J., Lacroix Y.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 3
193-213
EN
A combinatorial description of spectral isomorphisms between Morse flows is provided. We introduce the notion of a regular spectral isomorphism and we study some invariants of such isomorphisms. In the case of Morse cocycles taking values in $G = ℤ_p$, where p is a prime, each spectral isomorphism is regular. The same holds true for arbitrary finite abelian groups under an additional combinatorial condition of asymmetry in the defining Morse sequence, and for Morse flows of rank one. Rank one is shown to be a spectral invariant in the class of Morse flows.
7
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Anzelm Iwanik (1946-1998)

26%
Downarowicz T., Lipecki Z.
Colloquium Mathematicum
|
2000
|
tom 84/85
|
nr 1
1-12
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