Homoclinic points of algebraic ℤ^{𝕕}-actions

Lind, Douglas ¹   ; Schmidt, Klaus ²

¹ Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195–4350
² Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria

Journal of the American Mathematical Society, Tome 12 (1999) no. 4, pp. 953-980

Résumé

Let $\alpha$ be an action of $\mathbb Z^d$ by continuous automorphisms of a compact abelian group $X$. A point $x$ in $X$ is called homoclinic for $\alpha$ if $\alpha ^{\mathbf n}x\to 0_X$ as $\|\mathbf n\|\to \infty$. We study the set $\Delta _{\alpha }(X)$ of homoclinic points for $\alpha$, which is a subgroup of $X$. If $\alpha$ is expansive, then $\Delta _{\alpha }(X)$ is at most countable. Our main results are that if $\alpha$ is expansive, then (1) $\Delta _{\alpha }(x)$ is nontrivial if and only if $\alpha$ has positive entropy and (2) $\Delta _{\alpha }(X)$ is nontrivial and dense in $X$ if and only if $\alpha$ has completely positive entropy. In many important cases $\Delta _{\alpha }(X)$ 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 $\mathbb Z^d$-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.

Détail

DOI : 10.1090/S0894-0347-99-00306-9

Bibliographie

[1] Arrowsmith, D. K., Place, C. M. An introduction to dynamical systems 1990

[2] Feller, William An introduction to probability theory and its applications. Vol. I 1968

[3] Kaminker, Jerome, Putnam, Ian 𝐾-theoretic duality of shifts of finite type Comm. Math. Phys. 1997 509 522

[4] Katok, Anatole B., Schmidt, Klaus The cohomology of expansive 𝑍^{𝑑}-actions by automorphisms of compact, abelian groups Pacific J. Math. 1995 105 142

[5] Katznelson, Yitzhak, Weiss, Benjamin Commuting measure-preserving transformations Israel J. Math. 1972 161 173

[6] Lind, D. A. Split skew products, a related functional equation, and specification Israel J. Math. 1978 236 254

[7] Lind, D. A. Ergodic group automorphisms and specification 1979 93 104

[8] Lind, Douglas, Schmidt, Klaus, Ward, Tom Mahler measure and entropy for commuting automorphisms of compact groups Invent. Math. 1990 593 629

[9] Ergodic theory, symbolic dynamics, and hyperbolic spaces 1991

[10] Parry, William, Tuncel, Selim Classification problems in ergodic theory 1982

[11] Putnam, Ian F. 𝐶*-algebras from Smale spaces Canad. J. Math. 1996 175 195

[12] Ruelle, David Statistical mechanics on a compact set with 𝑍^{𝑣} action satisfying expansiveness and specification Trans. Amer. Math. Soc. 1973 237 251

[13] Schmidt, Klaus Automorphisms of compact abelian groups and affine varieties Proc. London Math. Soc. (3) 1990 480 496

[14] Schmidt, Klaus Dynamical systems of algebraic origin 1995

[15] Schmidt, K. Cohomological rigidity of algebraic 𝐙^{𝐝}-actions Ergodic Theory Dynam. Systems 1995 759 805

[16] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals 1993

[17] Wolfart, Jürgen Werte hypergeometrischer Funktionen Invent. Math. 1988 187 216