Diffeomorphisms on the Torus

For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior. A diffeomorphism is Morse-Smale provided it is structurally stable and has a finite nonwandering set [P-S]. Several recent papers have explored the relationship between the topology of these maps and their dynamics. In [F] Franks showed that the periodic behavior of a Morse-Smale diffeomorphism was restricted by its homology zeta function. For the homotopy class of the identity on a compact surface Narasimhan proved that virtually any periodic behavior consistent with the homology zeta function does indeed occur [N]. For orientation-reversing maps there are obstructions other than the homology zeta function. Blanchard and Franks [B-F] have shown that if an orientation-reversing homeomorphism of S2 has periodic orbits which include two distinct odd periods, then the entropy of that map is positive. This implies that no orientationreversing Morse-Smale diffeomorphism on S2 can have distinct odd periods. The following theorem was conjectured in [B-F] and has been proven by Handel. Theorem [H]. Iff: M2 —> M2 is an orientation-reversing homeomorphism of a compact oriented surface of genus g, and iff has orbits with g + 2 distinct odd periods, then the entropy off is positive. Thus an orientation-reversing Morse-Smale diffeomorphism on the torus has orbits with at most two different odd periods. In this paper we investigate whether there are any further restrictions on its periodic behavior. In [B] we showed that it suffices to consider the isotopy classes of the toral diffeomorphisms induced by (¿ _J) and (¿ _¡). We will show that there is a further obstruction in the former class but not in the latter. I would like to thank John Franks and Lynn Narasimhan for their contributions to this paper. Preliminaries. In this section we state our main result following some necessary definitions and background. We assume the reader is familiar with various standard terms and notation from dynamical systems. Further details are available in [B], [N] and [Ni]. _ Received by the editors November 26, 1979 and, in revised form, February 25, 1980. AMS (MOS) subject classifications (1970). Primary 58F09, 58F20. "Research supported in part by Emory University Summer Research Grant. © 1981 American Mathematical Society 0002-9947/81/0000-0101/$03.2 S 29 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use