Automorphisms with finite exact uniform rank

Mieczysław K. Mentzen

ArticleOriginal scientific text

Title

Automorphisms with finite exact uniform rank

Authors ¹

Affiliations

Institute of Mathematics, Nicolas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Abstract

The notion of exact uniform rank, EUR, of an automorphism of a probability Lebesgue space is defined. It is shown that each ergodic automorphism with finite EUR is finite extension of some automorphism with rational discrete spectrum. Moreover, for automorphisms with finite EUR, the upper bounds of EUR of their factors and ergodic iterations are computed.

F. M. Dekking, Combinatorial and statistical properties of sequences generated by substitutions, thesis, 1980.
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrsch. Verw. Gebiete 41 (1978), 221-239.
A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math. 29 (3) (1977), 655-633.
A. del Junco, Transformations with discrete spectra are stacking transformations, ibid. 28 (1976), 836-839.
J. King, For mixing transformations rank $(T^{k}) = k \cdot r a n k (T)$ , Israel J. Math. 56 (1986), 102-122.
M. Lemańczyk and M. K. Mentzen, on metric properties of substitutions, Compositio Math. 65 (1988), 241-263.
D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. (2) 19 (1979), 129-136.
D. Ornstein, D. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 37 (262) (1982).
M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987.
V. A. Rokhlin, On fundamental ideas in measure theory, Mat. Sb. 25 (67) (1) (1949), 107-150 (in Russian)

Title

Automorphisms with finite exact uniform rank

Affiliations

Abstract

Bibliography