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Ergodic properties of generalized Lüroth series. (English) Zbl 0848.11039

Lüroth series belong to classical expansions of real numbers. If \(x\in (0,1]\) then \(x\) can be written in the form \(x= {1\over a_1}+ {1\over s_1 a_1}+\cdots+ {1\over s_1\cdots s_{n- 1} a_n}+\cdots\), where \(a_k\geq 2\) are integers, \(s_k= a_k\) \((a_k- 1)\) \((k= 1, 2,\dots)\) [cf. O. Perron, Irrationalzahlen, de Gruyter, Berlin-Leipzig (1921; JFM 48.0190.05), 116-122]. The metric theory of these expansions has been built in the second half of the 20-th century. The Lüroth expansions are based on the transformation \(T_L: [0, 1]\to [0, 1]\), \(T_L 0= 0\), \(T_L x= [{1\over x}] ([{1\over x}]+ 1) \cdot x- [{1\over x}]\) for \(x\in (0, 1]\) (\([t]\) is the greatest integer not exceeding \(t\)).
In this paper, the authors construct generalized Lüroth series (GLS) using two transformations \(T\), \(S\) defined by the help of a system of intervals \(I_n= (\ell_n, r_n]\) \((n\in D\subset N)\) such that \(\sum_{n\in D} (r_n- \ell_n)= 1\). The functions \(T\), \(S\) are linear on \(I_n\) \((n\in D)\) and equal 0 on the complement of \(\bigcup_{n\in D} I_n\). The authors also investigate ergodic properties of GLS and study the relation between GLS-transformations and \(\beta\)-transformations \(T_\beta\), where \(T_\beta x= \beta x\text{ mod } 1\), \(\beta> 1\), \(\beta\) irrational.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28D05 Measure-preserving transformations

Citations:

JFM 48.0190.05
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