On the multiplicity function of ergodic group extensions, II

Kwiatkowski, Jakub ; Lemańczyk, Mariusz

Studia Mathematica, Tome 113 (1995), p. 207-215 / Harvested from The Polish Digital Mathematics Library

Résumé

For an arbitrary set $A ⊆ ℕ^+$ containing 1, an ergodic automorphism T whose set of essential values of the multiplicity function is equal to A is constructed. If A is additionally finite, T can be chosen to be an analytic diffeomorphism on a finite-dimensional torus.

Détail

Publié le : 1995-01-01

Zbl 0857.28012

EUDML-ID : urn:eudml:doc:216228

Bibliographie

[00000] [1] L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk SSSR 26 (1962), 513-550 (in Russian).

[00001] [2] O. N. Ageev, Dynamical systems with a Lebesgue component of even multiplicity in the spectrum, Mat. Sb. 136 (178) (1988), 307-319 (in Russian).

[00002] [3] F. Blanchard and M. Lemańczyk, Measure preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal. 1 (1993), 275-294. | Zbl 0793.28012

[00003] [4] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1981.

[00004] [5] S. Ferenczi and J. Kwiatkowski, Rank and spectral multiplicity, Studia Math. 102 (1992), 121-144. | Zbl 0809.28013

[00005] [6] S. Ferenczi, J. Kwiatkowski and C. Mauduit, Density theorem for (multiplicity, rank) pairs, J. Anal. Math., to appear. | Zbl 0833.28010

[00006] [7] G. R. Goodson, On the spectral multiplicity of a class of finite rank transformations, Proc. Amer. Math. Soc. 93 (1985), 303-306. | Zbl 0551.28019

[00007] [8] G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet, On the multiplicity function of ergodic group extensions of rotations, Studia Math. 102 (1992), 157-174. | Zbl 0830.28009

[00008] [9] G. R. Goodson and M. Lemańczyk, On the rank of a class of bijective substitutions, ibid. 96 (1990), 219-230. | Zbl 0711.28007

[00009] [10] A. del Junco and D. Rudolph, Simple rigid rank-1, Ergodic Theory Dynam. Systems 7 (1987), 229-247. | Zbl 0634.54020

[00010] [11] A. B. Katok, Constructions in ergodic theory, preprint. | Zbl 1030.37001

[00011] [12] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (5) (1967), 81-106 (in Russian); English transl.: Russian Math. Surveys 15 (1967), 1-22. | Zbl 0172.07202

[00012] [13] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math. 51 (1988), 182-227. | Zbl 0665.28010

[00013] [14] J. Kwiatkowski and A. Sikorski, Spectral properties of G-symbolic Morse shifts, Bull. Soc. Math. France 115 (1987), 19-33. | Zbl 0624.28014

[00014] [15] M. Lemańczyk, Toeplitz $Z_2$-extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43.

[00015] [16] 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

[00016] [17] M. K. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. 35 (1987), 417-424. | Zbl 0675.28006

[00017] [18] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 776-779 (in Russian). | Zbl 0152.33404

[00018] [19] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, 1981. | Zbl 0449.28016

[00019] [20] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer 1987. | Zbl 0642.28013

[00020] [21] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. | Zbl 0519.28008

[00021] [22] E. A. Robinson, Transformations with highly nonhomogeneous spectrum of finite multiplicity, Israel J. Math. 56 (1986), 75-88. | Zbl 0614.28012

[00022] [23] E. A. Robinson, Spectral multiplicity for non-abelian Morse sequences, in: Lecture Notes in Math. 1342, Springer, 1988, 645-652.