Constructing Smale diffeomorphisms on compact surfaces

Batterson, Steve

doi:10.1090/S0002-9947-1980-0549164-1

Constructing Smale diffeomorphisms on compact surfaces
HTML articles powered by AMS MathViewer

by Steve Batterson PDF

Trans. Amer. Math. Soc. 257 (1980), 237-245 Request permission

Abstract:

A necessary condition for an isotopy class on a compact surface to admit a Smale diffeomorphism whose dynamics are a specified set of subshifts of finite type is that the Euler characteristic of the manifold be equal to a sum and difference of certain numbers obtained from the matrices representing the subshifts. In this paper it is shown that this condition is sufficient up to a finite power of the subshifts.

References

Rufus Bowen, Topological entropy and axiom $\textrm {A}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR 0262459
R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 43–49. MR 0271401
John M. Franks, Constructing structurally stable diffeomorphisms, Ann. of Math. (2) 105 (1977), no. 2, 343–359. MR 448434, DOI 10.2307/1971001
John M. Franks, Non-singular flows on $S^{3}$ with hyperbolic chain-recurrent set, Rocky Mountain J. Math. 7 (1977), no. 3, 539–546. MR 488164, DOI 10.1216/RMJ-1977-7-3-539

Algebraic topology

Tameness and stability

J. W. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447–493. MR 287580, DOI 10.2307/1970766
Clark Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28–73. MR 474411, DOI 10.1016/0022-0396(76)90004-8
Michael Shub, Structurally stable diffeomorphisms are dense, Bull. Amer. Math. Soc. 78 (1972), 817–818. MR 307278, DOI 10.1090/S0002-9904-1972-13047-7
M. Shub and D. Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109–132. MR 400306, DOI 10.1016/0040-9383(75)90022-1
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1

Stability and isotopy in discrete dynamical systems

R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. MR 331436, DOI 10.2307/1970908