Markov partitions for the two-dimensional torus
| Autor: | Mark R. Snavely |
|---|---|
| Rok vydání: | 1991 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 113:517-527 |
| ISSN: | 1088-6826 0002-9939 |
| Popis: | We examine Markov partitions for hyperbolic automorphisms of T2 in the spirit of Adler, Weiss, and others and give necessary conditions on the transition matrix of a Markov partition for a given automorphism. We give necessary and sufficient conditions for partitions with two connected rectangles. 1. BACKGROUND DEFINITIONS We begin by briefly reviewing the notions of a hyperbolic automorphism of 22 2/Z2 27r 2Z T (TI =R /2 where Ri -* Ri /2) and that of a Markov partition. For a more detailed introduction see [AW, D, Sn]. Let v be a 2 x 2 matrix with integer entries. Suppose also that det(V) = ?1 so that v 1 is an integer matrix. In other words, v E GL(2, 2). We further suppose that none of the eigenvalues of v have modulus 1. Therefore by the Perron-Froebenius Theorem we have real eigenvalues Au and As satisfying KIAuI > 1 > IAs > 0. We call Au the unstable eigenvalue of v and As the stable eigenvalue of v . The corresponding eigenvectors, V'U and V', are called the unstable and stable eigenvectors respectively. We note at this time that Tr(sV) 0 0. If Tr(sV) > 0, then Au > 0 and if Tr(Q) < 0, then Au < 0. v induces an automorphism of Tj2 , which we will also denote X. We define Wu(x) for x E Tjp as the projection of a line through K lx parallel to V'U with Ws (x) defined similarly. Definition 1.1. We define D = {Ri}l= to be a Markov partition for v if the following are true: (i) T2 =U7n1,Ri i , (ii) Ri = interior(Rd); (iii) interior(Ri) n interior(Rj) = 0, 1i 0 j; (iv) if x, y E Ri, then Wu(x, R,) n Ws(y, R1) = z E Ri (the intersection is a single point z); (v) V (Wu(x, R1)) D Wu(,Vx, RJ) where x E Ri and SVx E R Received by the editors December 5, 1989; the contents of this paper were presented at the Conference/Workshop on Ergodic Theory and Symbolic Dynamics, University of Washington, Seattle, on June 21, 1989. This conference was sponsored in part by the NSF. 1980 Mathematics Subject Classification (1985 Revision). Primary 58Fl 5. ? 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|