For a dynamical system (X,T,μ), we investigate the connections between a metric invariant, the rank r(T), and a spectral invariant, the maximal multiplicity m(T). We build examples of systems for which the pair (m(T),r(T)) takes values (m,m) for any integer m ≥ 1 or (p-1, p) for any prime number p ≥ 3.
@article{bwmeta1.element.bwnjournal-article-smv102i2p121bwm,
author = {S\'ebastien Ferenczi and Jan Kwiatkowski},
title = {Rank and spectral multiplicity},
journal = {Studia Mathematica},
volume = {103},
year = {1992},
pages = {121-144},
zbl = {0809.28013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv102i2p121bwm}
}
Ferenczi, Sébastien; Kwiatkowski, Jan. Rank and spectral multiplicity. Studia Mathematica, Tome 103 (1992) pp. 121-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv102i2p121bwm/
[00000] [Age] O. N. Ageev, Dynamical systems with a Lebesgue component of even multiplicity in the spectrum, Mat. Sb. 136 (178) (1988), 307-319 (in Russian).
[00001] [Cha1] R. V. Chacon, A geometric construction of measure preserving transformations, in: Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part 2, Univ. of California Press, 1965, 335-360.
[00002] [Cha2] R. V. Chacon, Approximation and spectral multiplicity, in: Contributions to Ergodic Theory and Probability, Lecture Notes in Math. 160, Springer, 1970, 18-27.
[00003] [Fer1] S. Ferenczi, Systèmes localement de rang un, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), 35-51. | Zbl 0535.28010
[00004] [Fer2] S. Ferenczi, Tiling and local rank properties of the Morse sequence, Theoret. Comput. Sci., to appear.
[00005] [GKLL] G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet, On the multiplicity function of ergodic group extensions of rotations, this volume, 157-174. | Zbl 0830.28009
[00006] [GoLe] G. R. Goodson and M. Lemańczyk, On the rank of a class of bijective substitutions, Studia Math. 96 (1990), 219-230. | Zbl 0711.28007
[00007] [delJ] A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math. 29 (1977), 655-663. | Zbl 0335.28010
[00008] [Kea1] M. Keane, Generalized Morse sequences, Z. Wahrsch. Verw. Gebiete 10 (1968), 335-353. | Zbl 0162.07201
[00009] [Kea2] M. Keane, Strongly mixing g-measures, Invent. Math. 16 (1972), 309-353. | Zbl 0241.28014
[00010] [Kwi] J. Kwiatkowski, Isomorphism of regular Morse dynamical systems, Studia Math. 72 (1982), 59-89. | Zbl 0525.28018
[00011] [KwRo] J. Kwiatkowski and T. Rojek, A method of solving a cocycle functional equation and applications, ibid. 99 (1991), 69-86. | Zbl 0734.28016
[00012] [KwSi] J. Kwiatkowski and A. Sikorski, Spectral properties of G-symbolic Morse shifts, Bull. Soc. Math. France 115 (1987), 19-33. | Zbl 0624.28014
[00013] [Lem] M. Lemańczyk, Toeplitz Z₂-extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43.
[00014] [Mar1] J. C. Martin, Generalized Morse sequences on n symbols, Proc. Amer. Math. Soc. 54 (1976), 379-383.
[00015] [Mar2] J. C. Martin, The structure of generalized Morse minimal sets on n symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355.
[00016] [MaNa] J. Mathew and M. G. Nadkarni, A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, Bull. London Math. Soc. 16 (1984), 402-406. | Zbl 0515.28010
[00017] [Men1] M. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. Math. 35 (1987), 417-424. | Zbl 0675.28006
[00018] [Men2] M. Mentzen, Thesis, preprint no. 2/89, Nicholas Copernicus University, Toruń 1989.
[00019] [New] D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. 19 (1979), 129-136. | Zbl 0425.28012
[00020] [ORW] D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 262 (1982). | Zbl 0504.28019
[00021] [Par] W. Parry, Compact abelian group extensions of discrete dynamical systems, Z. Wahrsch. Verw. Gebiete 13 (1969), 95-113. | Zbl 0184.26901
[00022] [Que] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987. | Zbl 0642.28013
[00023] [Rob1] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. | Zbl 0519.28008
[00024] [Rob2] E. A. Robinson, Mixing and spectral multiplicity, Ergodic Theory Dynamical Systems 5 (1985), 617-624. | Zbl 0565.28013