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1
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Two commuting maps without common minimal points

100%
Downarowicz T.
Colloquium Mathematicum
|
2011
|
tom 123
|
nr 2
205-209
EN
We construct an example of two commuting homeomorphisms S, T of a compact metric space X such that the union of all minimal sets for S is disjoint from the union of all minimal sets for T. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of "doubly recurrent" points (see [F]). Not only are these points recurrent under both S and T, but they recur along the same sequence of powers. Our example shows that nothing similar holds if recurrence is replaced by the stronger notion of uniform recurrence.
2
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Large sets of integers and hierarchy of mixing properties of measure preserving systems

64%
Bergelson V., Downarowicz T.
Colloquium Mathematicum
|
2008
|
tom 110
|
nr 1
117-150
EN
We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R^{ε}_{A,B} = {n ∈ ℤ: μ(A ∩ TⁿB) > μ(A)μ(B) - ε}$ of times of "fat intersection". Among other things we show that the sets $R^{ε}_{A,B}$ allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary ℤ-actions.
3
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Epsilon-independence between two processes

64%
Downarowicz T., Grzegorek P.
Studia Mathematica
|
2008
|
tom 188
|
nr 1
77-95
EN
We study the notion of ε-independence of a process on finitely (or countably) many states and that of ε-independence between two processes defined on the same measure preserving transformation. For that we use the language of entropy. First we demonstrate that if a process is ε-independent then its ε-independence from another process can be verified using a simplified condition. The main direction of our study is to find natural examples of ε-independence. In case of ε-independence of one process, we find an example among processes generated on the induced (first return time) transformation defined on a typical long cylinder set of any given process of positive entropy. To obtain examples of pairs of ε-independent processes we have to make an additional assumption on the master process. Then again, we find such pairs generated on the induced transformation as above. This is the most elaborate part of the paper. While the question whether our assumption is necessary remains open, we indicate a large class of processes where our assumption is satisfied.
4
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Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

64%
Downarowicz T., Kasjan S.
Studia Mathematica
|
2015
|
tom 229
|
nr 1
45-72
EN
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].
5
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Phenomena in rank-one ℤ²-actions

64%
Downarowicz T., Serafin J.
Studia Mathematica
|
2009
|
tom 192
|
nr 3
281-294
EN
We present an example of a rank-one partially mixing ℤ²-action which possesses a non-rigid factor and for which the Weak Closure Theorem fails. This is in sharp contrast to one-dimensional actions, which cannot display this type of behavior.
6
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Faithful zero-dimensional principal extensions

64%
Downarowicz T., Huczek D.
Studia Mathematica
|
2012
|
tom 212
|
nr 1
1-19
EN
We prove that every topological dynamical system (X,T) has a faithful zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν | X) is zero, and, in addition, every invariant measure on X has exactly one preimage on Y. This is a strengthening of the authors' result in Acta Appl. Math. [to appear] (where the extension was principal, but not necessarily faithful).
7
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When every point is either transitive or periodic

64%
Downarowicz T., Ye X.
Colloquium Mathematicum
|
2002
|
tom 93
|
nr 1
137-150
EN
We study transitive non-minimal ℕ-actions and ℤ-actions. We show that there are such actions whose non-transitive points are periodic and whose topological entropy is positive. It turns out that such actions can be obtained by perturbing minimal systems under some reasonable assumptions.
8
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Fiber entropy and conditional variational principles in compact non-metrizable spaces

64%
Downarowicz T., Serafin J.
Fundamenta Mathematicae
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2002
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tom 172
|
nr 3
217-247
EN
We consider a pair of topological dynamical systems on compact Hausdorff (not necessarily metrizable) spaces, one being a factor of the other. Measure-theoretic and topological notions of fiber entropy and conditional entropy are defined and studied. Abramov and Rokhlin's definition of fiber entropy is extended, using disintegration. We prove three variational principles of conditional nature, partly generalizing some results known before in metric spaces: (1) the topological conditional entropy equals the supremum of the topological fiber entropy over the factor, which also equals the supremum of the topological fiber entropy given a measure over all invariant measures on the factor, (2) the topological fiber entropy given a measure equals the supremum of the measure-theoretic conditional entropy over all invariant measures on the larger system projecting to the given one. Combining the above, we get (3) the topological conditional entropy equals the supremum of the measure-theoretic conditional entropy over all invariant measures. A tail entropy of a measure is introduced in totally disconnected spaces. As an application of our variational principles it is proved that the tail entropy estimates from below the "defect of upper semicontinuity" of the entropy function.
9
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Dynamiques associees a une echelle de numeration

52%
Barat G., Downarowicz T., Liardet P.
Acta Arithmetica
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2002
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tom 103
|
nr 1
41-78
10
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Propriétés topologiques et combinatoires des échelles de numération

52%
Barat G., Downarowicz T., Iwanik A., Liardet P.
Colloquium Mathematicum
|
2000
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tom 84/85
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nr 2
285-306
EN
Topological and combinatorial properties of dynamical systems called odometers and arising from number systems are investigated. First, a topological classification is obtained. Then a rooted tree describing the carries in the addition of 1 is introduced and extensively studied. It yields a description of points of discontinuity and a notion of low scale, which is helpful in producing examples of what the dynamics of an odometer can look like. Density of the orbits is also discussed.
11
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Some properties of weakly almost periodic mappings on compact spaces

50%
Downarowicz T.
Colloquium Mathematicum
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1987
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tom 54
|
nr 2
241-252
12
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Multiple recurrence for discrete time Markov processes. II

45%
Downarowicz T., Iwanik A.
Colloquium Mathematicum
|
1988
|
tom 55
|
nr 2
311-316
13
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Weakly almost periodic mappings on one- and two-manifolds

44%
Downarowicz T.
Colloquium Mathematicum
|
1987
|
tom 54
|
nr 2
253-259
14
Content available remote

Quasi-uniform convergence in compact dynamical systems

38%
Downarowicz T., Iwanik A.
Studia Mathematica
|
1988
|
tom 89
|
nr 1
11-25
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