Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T06:39:11.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic points and invariant pseudomeasures for toral endomorphisms

Published online by Cambridge University Press:  19 September 2008

William A. Veech
William A. Veech
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77251, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Extending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 3 , September 1986 , pp. 449 - 473
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Bary, N.. A Treatise on Trigonometric Series, Vol. II. Oxford, Pergamon Press, 1964.Google Scholar
[2]Cassels, J. W. S. & Frohlich, A., Eds. Algebraic Number Theory. Washington, D.C., Thompson Book Co., 1967.Google Scholar
[3]Chevalley, C.. Deux theoremes d'Arithmetique. J. Math. Soc. of Japan, 3 (1951), 36–44.CrossRefGoogle Scholar
[4]Davenport, H.. On some infinite series involving arithmetical functions. The Quarterly Journal of Mathematics, 8 (1937), 8–13.CrossRefGoogle Scholar
[5]Davenport, H.. On some infinite series involving arithmetical functions, (II). The Quarterly Journal of Mathematics, 8 (1937), 313–320.CrossRefGoogle Scholar
[6]Katznelson, Y.. Ergodic automorphisms on n are Bernoulli shifts. Israel J. Math. 10 (1971), 186–195.CrossRefGoogle Scholar
[7]Landau, E.. Handbuch über die Lehre von der Verteilung der Prinzahlen. Leipzig, Teubner, 1909.Google Scholar
[8]Lind, D., Dynamical properties of quasihyperbolic toral automorphisms. Ergod. Th. & Dynam. Sys. 2 (1982), 49–86.CrossRefGoogle Scholar
[9]Livšic, A. N.. Homology properties of Y systems. Math. Notes, 10 (1971), 758–763.Google Scholar
[10]Livšic, A. N.. Cohomology of dynamical systems. Math. U.S.S.R. Izvestija, 6 (1972), 1278–1301.CrossRefGoogle Scholar
[11]Marcus, B.. A note on periodic points for ergodic toral automorphisms. Monatsh. Math. 89 (1980), 121–129.CrossRefGoogle Scholar
[12]Stein, E. M.. Singular Integrals and Differentiability Properties of Functions. Princeton, Princeton University Press, 1970.Google Scholar
[13]Stein, E. M. & Weiss, G.. Introduction to Fourier Analysis on Euclidean Space. Princeton, Princeton University Press, 1971.Google Scholar
[14]Weiss, E.. Algebraic Number Theory. New York, McGraw-Hill, 1963.Google Scholar
[15]Weyl, H.. über die Gleichverteilung von Zahlen mod Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
[16]Zygmund, A.. Trigonometric Series, Vols. I, II. Cambridge, Cambridge University Press, 1959.Google Scholar