Automorphisms of hyperbolic dynamical systems and $K_ 2$
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- by Frank Zizza PDF
- Trans. Amer. Math. Soc. 307 (1988), 773-797 Request permission
Abstract:
Let $\sigma :\Sigma \to \Sigma$ be a subshift of finite type and $\operatorname {Aut} (\sigma )$ be the group of homeomorphisms of $\Sigma$ which commute with $\sigma$. In [Wl], Wagoner constructs an invariant for the group $\operatorname {Aut} (\sigma )$ using $K$-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner’s construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least $\bmod 2$) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.References
- Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
- Rufus Bowen, Entropy versus homology for certain diffeomorphisms, Topology 13 (1974), 61–67. MR 345135, DOI 10.1016/0040-9383(74)90038-X
- Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
- Mike Boyle, Douglas Lind, and Daniel Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc. 306 (1988), no. 1, 71–114. MR 927684, DOI 10.1090/S0002-9947-1988-0927684-2
- John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C. by the American Mathematical Society, Providence, R.I., 1982. MR 669378, DOI 10.1090/cbms/049
- M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, DOI 10.1007/BF01404552
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112 J. Wagoner, Markov partitions and ${K_2}$, Publ. Math. IHES, No. 65 (to appear).
- J. B. Wagoner, Realizing symmetries of a subshift of finite type by homeomorphisms of spheres, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 301–303. MR 828831, DOI 10.1090/S0273-0979-1986-15449-2
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 307 (1988), 773-797
- MSC: Primary 58F15; Secondary 19B99, 19C99, 54H20
- DOI: https://doi.org/10.1090/S0002-9947-1988-0940227-2
- MathSciNet review: 940227