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Maličky-Riečan's entropy as a version of operator entropy

100%
Frej B.
Fundamenta Mathematicae
|
2006
|
tom 189
|
nr 2
185-193
EN
The paper deals with the notion of entropy for doubly stochastic operators. It is shown that the entropy defined by Maličky and Riečan in [MR] is equal to the operator entropy proposed in [DF]. Moreover, some continuity properties of the [MR] entropy are established.
2
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A note on Markov operators and transition systems

100%
Frej B.
Colloquium Mathematicum
|
2002
|
tom 91
|
nr 2
183-190
EN
On a compact metric space X one defines a transition system to be a lower semicontinuous map $X → 2^{X}$. It is known that every Markov operator on C(X) induces a transition system on X and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may not be induced by commuting Markov operators. The existence of trajectories for a pair of transition systems or Markov operators is also investigated.
3
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Minimal models for $ℤ^{d}$-actions

64%
Frej B., Kwaśnicka A.
Colloquium Mathematicum
|
2008
|
tom 110
|
nr 2
461-476
EN
We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$-action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$-action with the same, i.e. affinely homeomorphic, simplex of measures.
4
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An integral formula for entropy of doubly stochastic operators

64%
Frej B., Frej P.
Fundamenta Mathematicae
|
2011
|
tom 213
|
nr 3
271-289
EN
A new formula for entropy of doubly stochastic operators is presented. It is also checked that this formula fulfills the axioms of the axiomatic definition of operator entropy, introduced in an earlier paper of Downarowicz and Frej. As an application of the formula the 'product rule' is obtained, i.e. it is shown that the entropy of a product is the sum of the entropies of the factors. Finally, the proof of continuity of the new 'static' entropy as a function of the measure is given.
5
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A map maintaining the orbits of a given $ℤ^{d}$-action

64%
Frej B., Kwaśnicka A.
Colloquium Mathematicum
|
2016
|
tom 143
|
nr 1
1-15
EN
Giordano et al. (2010) showed that every minimal free $ℤ^{d}$-action of a Cantor space X is orbit equivalent to some ℤ-action. Trying to avoid the K-theory used there and modifying Forrest's (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map F on X∖{one point} such that for a residual subset of X the orbits of F are the same as the orbits of a given minimal free $ℤ^{d}$-action.
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