Automorphisms of hyperbolic dynamical systems and

Author:
Frank Zizza

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subshift of finite type and be the group of homeomorphisms of which commute with . In [**Wl**], Wagoner constructs an invariant for the group using -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 ) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.

**[B1]**R. Bowen,*Markov partitions for Axiom A diffeomorphisms*, Amer. J. Math.**92**(1970), 725-747. MR**0277003 (43:2740)****[B2]**-,*Entropy versus homology for certain diffeomorphisms*, Topology**13**(1974), 61-67. MR**0345135 (49:9874)****[BF]**R. Bowen and J. Franks,*Homology for zero dimensional basic sets*, Ann. of Math. (2)**106**(1977), 72-92. MR**0458492 (56:16692)****[BLR]**M. Boyle, D. Lind and D. Rudolph,*The automorphism group of a subshift of finite type*, Trans. Amer. Math. Soc.**306**(1988), 71-114. MR**927684 (89m:54051)****[F]**J. Franks,*Homology and dynamical systems*, CBMS Regional Conf. Ser. in Math., no. 49, Amer. Math. Soc., Providence, R. I., 1982. MR**669378 (84f:58067)****[HPPS]**M. Hirsch, J. Palis, C. Pugh and M. Shub,*Neighborhoods of hyperbolic sets*, Invent. Math.**9**(1970), 121-134. MR**0262627 (41:7232)****[L]**S. Lang,*Algebra*, Addison-Wesley, Reading, Mass., 1965. MR**0197234 (33:5416)****[Sp]**E. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)****[W1]**J. Wagoner,*Markov partitions and*, Publ. Math. IHES, No. 65 (to appear).**[W2]**-,*Realizing symmetries of a subshift of finite type by homeomorphisms of spheres*, Bull. Amer. Math. Soc. (N.S.)**14**(1986), 301-303. MR**828831 (87f:58079)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F15,
19B99,
19C99,
54H20

Retrieve articles in all journals with MSC: 58F15, 19B99, 19C99, 54H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940227-2

Article copyright:
© Copyright 1988
American Mathematical Society