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Product of two staircase rank one transformations that is not loosely Bernoulli. (English) Zbl 1481.37004

Let \(r=(r_n)_{n \in \mathbb{N} }\) and \(s=((s_{n,j})_{j=0}^{p_n-1})\) be sequences of positive integers. The rank-one map with cutting and spacers parameters \((r,s)\) can be defined as the shift \(T\) on the set \(X\) of doubly infinite sequence of \(0\) and \(1\) obtained by concatenation of blocks by the rule \(B_0 = 0\), \(B_{n+1} = B_n1^{s_{n,1}}B_n1^{s_{n,2}}B_n\cdots B_n1^{s_{n,r_n}}\). The language associated with \(T\) is the set of words contained in some block \(B_n\). This is the constructive symbolic. The classical Chacon map is given by \(r_n=3\) and \(s_{n,1}=s_{n,3}=0, s_{n,2}=1\) for all \(n \geq 1\) and the staircase is given by \(s_{n,j}=j-1\), \(j=1, \dots, r_n-1\). The Ornstein random construction can be obtained by choosing randomly and uniformly \(x_{n,j}, j=1,\dots,r_n-1\) in the set \(\{1,\dots,S_n\}\), where \((S_n)\) is a fixed sequence of integers. Let \(s_{n,j}=S_n+x_{n,j}-x_{n,j-1},\) \(j=1,\dots,r_n\), \((x_{n,r_n})\) be a deterministic sequence.
The loosely Bernoulli is related to the metric \(\overline{f}\) introduced by J. Feldman [Isr. J. Math. 24, 16–38 (1976; Zbl 0336.28003)]. For two finite words (over a finite alphabet) \(A=a_1\cdots a_k\) and \(B=b_1 \cdots b_k \), a matching between \(A\) and \(B\) is any pair of strictly increasing sequences \((i_s,j_s )_ {s=1}^{r} \) such that \(a_{i_s}=b_{j_s}\) for \(s=1,\dots,r\). The \(\overline{f}\) distance between \(A\) and \(B\) is defined by \[\overline{f}(A,B)=1-\frac{r}{k}.\] where \(r\) is the maximal cardinality over all matchings between \(A\) and \(B\). Let \(\tilde{T}\) be a measure-preserving transformation on \((X,\mu),\) \(\mu\) is a probability measure. For a finite partition \(P=(P_1,\dots, P_r )\) of \(X\) and an integer \(N > 1\) we denote \(P_0^N(x) = x_0\cdots x_{N-1},\) where \(x_i \in \{1,\dots,r\}\) is such that \(T^i(x) \in P_{x_i}\) for \(i=0,\dots,N-1\). We recall that the zero entropy process \((\tilde{T},P)\) is said to be loosely Bernoulli if for every \(\varepsilon>0\) there exists \(N_{\varepsilon} \in \mathbb{N}\) and a measurable set \(A_{\varepsilon}\) with \(\mu(A_\varepsilon)>1- \varepsilon\), such that for every \(x, y \in A_{\varepsilon}\) and every \(N > N_{\varepsilon}\) \[\overline{f}(P_0^N(x),P_0^N(y))<\varepsilon.\]
It is well known that the rank-one maps are loosely Bernoulli and there is a rank-one map for which the Cartesian product is not loosely Bernoulli. This is due to D. S. Ornstein et al. [Equivalence of measure preserving transformations. Providence, RI: American Mathematical Society (AMS) (1982; Zbl 0504.28019)] and the construction is in the class of Ornstein random construction. Moreover, the spacers can be chosen such that the map is mixing, and the construction can be even extended to produce a loosely Bernoulli K-automorphism for which the Cartesian product is not loosely Bernoulli.
In this paper the authors investigate the property of loosely Bernoulli for the Cartesian product in the subclass of staircase rank-one maps. They further motivate their investigation by the open problem whether the Cartesian of Chacon map is loosely Bernoulli or not. This problem is due to A. Del Junco et al. [J. Anal. Math. 37, 276–284 (1980; Zbl 0445.28014)] but the authors erroneously attribute it to Jean-Paul Thouvenot.

MSC:

37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
37B10 Symbolic dynamics
28D05 Measure-preserving transformations
37H12 Random iteration
03C07 Basic properties of first-order languages and structures
68Q70 Algebraic theory of languages and automata
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References:

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