Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-06T14:15:48.031Z Has data issue: false hasContentIssue false
  • English
  • Français

Ledrappier’s system is almost mixing of all orders

Published online by Cambridge University Press:  01 April 2008

L. ARENAS-CARMONA ,
D. BEREND  and
V. BERGELSON
L. ARENAS-CARMONA
Affiliation:
Department of Mathematics, University of Chile, Casilla 653, Santiago, Chile (email: learenas@uchile.cl)
D. BEREND
Affiliation:
Departments of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel (email: berend@math.bgu.ac.il)
V. BERGELSON
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA (email: vitaly@math.ohio-state.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider Ledrappier’s dynamical system, which was the first example of a -action which is 2-mixing but not 3-mixing. Our main result is that, excluding certain small ‘constructible’ sets, the system is mixing of every order.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 339 - 365
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Einsiedler, M. and Ward, T.. Asymptotic geometry of non-mixing sequences. Ergod. Th. & Dynam. Sys. 23 (2003), 75–85.CrossRefGoogle Scholar
[2]Evertse, J.-H., Schlickewei, H. P. and Schmidt, W. M.. Linear equations in variables which lie in a multiplicative group. Ann. of Math. (2) 155 (2002), 807–836.CrossRefGoogle Scholar
[3]Ledrappier, F.. Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris 287 (1978), 561–563.Google Scholar
[4]Luca, F.. The Diophantine equation x 2=p a±p b+1. Acta Arith. 112 (2004), 87–101.CrossRefGoogle Scholar
[5]Masser, D. W.. Mixing and linear equations over groups in positive characteristic. Israel J. Math. 142 (2004), 189–204.CrossRefGoogle Scholar
[6]Rohlin, V. A.. On endomorphisms of compact commutative groups (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 13 (1949), 329–340.Google Scholar
[7]Schlickewei, H. P.. S-unit equations over number fields. Invent. Math. 102 (1990), 95–107.CrossRefGoogle Scholar
[8]Schmidt, K.. Dynamical Systems of Algebraic Origin. Birkhäuser, Basel, 1995.Google Scholar
[9]Schmidt, K. and Ward, T.. Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111 (1993), 69–76.CrossRefGoogle Scholar
[10]Scott, R.. Elementary treatment of p a±p b+1=x 2. http://arxiv.org/pdf/math/0608796 (also available at http://www.homepage.villanova.edu/robert.styer/ReeseScott/index.htm).Google Scholar
[11]Shorey, T. N. and Tijdeman, R.. Exponential Diophantine Equations. Cambridge University Press, Cambridge, 1986.CrossRefGoogle Scholar
[12]Szalay, L.. The equations 2N±2M±2L=z 2. Indag. Math. (N.S.) 13 (2002), 131–142.CrossRefGoogle Scholar
[13]Voloch, J. F.. The equation ax+by=1 in characteristic p. J. Number Theory 73 (1998), 195–200.CrossRefGoogle Scholar
[14]Weiss, B.. Personal communication.Google Scholar