Homoclinic points of algebraic -actions

Authors:
Douglas Lind and Klaus Schmidt

Journal:
J. Amer. Math. Soc. **12** (1999), 953-980

MSC (1991):
Primary 22D40, 54H20, 58F15; Secondary 13C10, 43A75

Published electronically:
May 24, 1999

MathSciNet review:
1678035

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an action of by continuous automorphisms of a compact abelian group . A point in is called homoclinic for if as . We study the set of homoclinic points for , which is a subgroup of . If is expansive, then is at most countable. Our main results are that if is expansive, then (1) is nontrivial if and only if has positive entropy and (2) is nontrivial and dense in if and only if has completely positive entropy. In many important cases is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of -actions to which Ruelle's thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.

**1.**D. K. Arrowsmith and C. M. Place,*An introduction to dynamical systems*, Cambridge University Press, Cambridge, 1990. MR**1069752****2.**William Feller,*An introduction to probability theory and its applications. Vol. I*, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0228020****3.**Jerome Kaminker and Ian Putnam,*𝐾-theoretic duality of shifts of finite type*, Comm. Math. Phys.**187**(1997), no. 3, 509–522. MR**1468312**, 10.1007/s002200050147**4.**Anatole B. Katok and Klaus Schmidt,*The cohomology of expansive 𝑍^{𝑑}-actions by automorphisms of compact, abelian groups*, Pacific J. Math.**170**(1995), no. 1, 105–142. MR**1359974****5.**Yitzhak Katznelson and Benjamin Weiss,*Commuting measure-preserving transformations*, Israel J. Math.**12**(1972), 161–173. MR**0316680****6.**D. A. Lind,*Split skew products, a related functional equation, and specification*, Israel J. Math.**30**(1978), no. 3, 236–254. MR**508267**, 10.1007/BF02761073**7.**D. A. Lind,*Ergodic group automorphisms and specification*, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 93–104. MR**550414****8.**Douglas Lind, Klaus Schmidt, and Tom Ward,*Mahler measure and entropy for commuting automorphisms of compact groups*, Invent. Math.**101**(1990), no. 3, 593–629. MR**1062797**, 10.1007/BF01231517**9.**Tim Bedford, Michael Keane, and Caroline Series (eds.),*Ergodic theory, symbolic dynamics, and hyperbolic spaces*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17–28, 1989. MR**1130170****10.**William Parry and Selim Tuncel,*Classification problems in ergodic theory*, London Mathematical Society Lecture Note Series, vol. 67, Cambridge University Press, Cambridge-New York, 1982. Statistics: Textbooks and Monographs, 41. MR**666871****11.**Ian F. Putnam,*𝐶*-algebras from Smale spaces*, Canad. J. Math.**48**(1996), no. 1, 175–195. MR**1382481**, 10.4153/CJM-1996-008-2**12.**David Ruelle,*Statistical mechanics on a compact set with 𝑍^{𝑣} action satisfying expansiveness and specification*, Trans. Amer. Math. Soc.**187**(1973), 237–251. MR**0417391**, 10.1090/S0002-9947-1973-0417391-6**13.**D. Ruelle,*Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics*, Addison-Wesley, Reading, MA, 1978.**14.**Klaus Schmidt,*Automorphisms of compact abelian groups and affine varieties*, Proc. London Math. Soc. (3)**61**(1990), no. 3, 480–496. MR**1069512**, 10.1112/plms/s3-61.3.480**15.**Klaus Schmidt,*Dynamical systems of algebraic origin*, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995. MR**1345152****16.**K. Schmidt,*Cohomological rigidity of algebraic 𝐙^{𝐝}-actions*, Ergodic Theory Dynam. Systems**15**(1995), no. 4, 759–805. MR**1346399**, 10.1017/S0143385700008646**17.**K. Schmidt,*On the cohomology of algebraic -actions with values in compact Lie groups*, Colloquium on Lie Groups and Ergodic Theory, Tata Institute, 1996 (to appear).**18.**Elias M. Stein,*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192****19.**Jürgen Wolfart,*Werte hypergeometrischer Funktionen*, Invent. Math.**92**(1988), no. 1, 187–216 (German, with English summary). MR**931211**, 10.1007/BF01393999

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
22D40,
54H20,
58F15,
13C10,
43A75

Retrieve articles in all journals with MSC (1991): 22D40, 54H20, 58F15, 13C10, 43A75

Additional Information

**Douglas Lind**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195–4350

Email:
lind@math.washington.edu

**Klaus Schmidt**

Affiliation:
Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria

Email:
klaus.schmidt@univie.ac.at

DOI:
https://doi.org/10.1090/S0894-0347-99-00306-9

Keywords:
Homoclinic point,
algebraic action,
expansive action,
specification

Received by editor(s):
February 13, 1997

Received by editor(s) in revised form:
May 30, 1998

Published electronically:
May 24, 1999

Additional Notes:
Both authors were supported in part by NSF Grant DMS-9303240. The first author was also supported in part by NSF Grant DMS-9622866.

Article copyright:
© Copyright 1999
American Mathematical Society