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1
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On strong uniform distribution, II. The infinite-dimensional case

100%
Lacroix Y.
Acta Arithmetica
|
1998
|
tom 84
|
nr 3
279-290
EN
We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^∞$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^∞$. There also exist such bad chains with zero density.
2
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Metric properties of generalized Cantor products

100%
Lacroix Y.
Acta Arithmetica
|
1993
|
tom 63
|
nr 1
61-77
3
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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.
4
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Some constructions of strictly ergodic non-regular Toeplitz flows

64%
Iwanik A., Lacroix Y.
Studia Mathematica
|
1994
|
tom 110
|
nr 2
191-203
EN
We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.
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
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1995
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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.
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