On the multiplicity function of ergodic group extensions, II

Jakub Kwiatkowski; Mariusz Lemańczyk

Studia Mathematica (1995)

  • Volume: 116, Issue: 3, page 207-215
  • ISSN: 0039-3223

Abstract

top
For an arbitrary set 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.

How to cite

top

Kwiatkowski, Jakub, and Lemańczyk, Mariusz. "On the multiplicity function of ergodic group extensions, II." Studia Mathematica 116.3 (1995): 207-215. <http://eudml.org/doc/216228>.

@article{Kwiatkowski1995,
abstract = {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.},
author = {Kwiatkowski, Jakub, Lemańczyk, Mariusz},
journal = {Studia Mathematica},
keywords = {ergodic group extensions; multiplicity function; Morse automorphisms; weakly mixing},
language = {eng},
number = {3},
pages = {207-215},
title = {On the multiplicity function of ergodic group extensions, II},
url = {http://eudml.org/doc/216228},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Kwiatkowski, Jakub
AU - Lemańczyk, Mariusz
TI - On the multiplicity function of ergodic group extensions, II
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 3
SP - 207
EP - 215
AB - 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.
LA - eng
KW - ergodic group extensions; multiplicity function; Morse automorphisms; weakly mixing
UR - http://eudml.org/doc/216228
ER -

References

top
  1. [1] L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk SSSR 26 (1962), 513-550 (in Russian). 
  2. [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). 
  3. [3] F. Blanchard and M. Lemańczyk, Measure preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal. 1 (1993), 275-294. Zbl0793.28012
  4. [4] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1981. 
  5. [5] S. Ferenczi and J. Kwiatkowski, Rank and spectral multiplicity, Studia Math. 102 (1992), 121-144. Zbl0809.28013
  6. [6] S. Ferenczi, J. Kwiatkowski and C. Mauduit, Density theorem for (multiplicity, rank) pairs, J. Anal. Math., to appear. Zbl0833.28010
  7. [7] G. R. Goodson, On the spectral multiplicity of a class of finite rank transformations, Proc. Amer. Math. Soc. 93 (1985), 303-306. Zbl0551.28019
  8. [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. Zbl0830.28009
  9. [9] G. R. Goodson and M. Lemańczyk, On the rank of a class of bijective substitutions, ibid. 96 (1990), 219-230. Zbl0711.28007
  10. [10] A. del Junco and D. Rudolph, Simple rigid rank-1, Ergodic Theory Dynam. Systems 7 (1987), 229-247. Zbl0634.54020
  11. [11] A. B. Katok, Constructions in ergodic theory, preprint. Zbl1030.37001
  12. [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. Zbl0172.07202
  13. [13] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math. 51 (1988), 182-227. Zbl0665.28010
  14. [14] J. Kwiatkowski and A. Sikorski, Spectral properties of G-symbolic Morse shifts, Bull. Soc. Math. France 115 (1987), 19-33. Zbl0624.28014
  15. [15] M. Lemańczyk, Toeplitz -extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43. 
  16. [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. Zbl0515.28010
  17. [17] M. K. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. 35 (1987), 417-424. Zbl0675.28006
  18. [18] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 776-779 (in Russian). Zbl0152.33404
  19. [19] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, 1981. Zbl0449.28016
  20. [20] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer 1987. Zbl0642.28013
  21. [21] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. Zbl0519.28008
  22. [22] E. A. Robinson, Transformations with highly nonhomogeneous spectrum of finite multiplicity, Israel J. Math. 56 (1986), 75-88. Zbl0614.28012
  23. [23] E. A. Robinson, Spectral multiplicity for non-abelian Morse sequences, in: Lecture Notes in Math. 1342, Springer, 1988, 645-652. 

NotesEmbed ?

top

You must be logged in to post comments.