Homology for one-dimensional solenoids
Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.
Aoki, N. and Hiraide, K., Topological theory of dynamical systems: Recent advances, North-Holland Mathematical Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994.
Bowen, R., Markov partitions for Axiom $rm A$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. https://doi.org/10.2307/2373370
Fisher, T., Resolving extensions of finitely presented systems, Acta Appl. Math. 126 (2013), 131–163. https://doi.org/10.1007/s10440-013-9811-x
Fried, D., Finitely presented dynamical systems, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 489–507. https://doi.org/10.1017/S014338570000417X
Lind, D. and Marcus, B., An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511626302
Putnam, I. F., A homology theory for Smale spaces, vol. 232, Mem. Amer. Math. Soc., no. 1094, Amer. Math. Soc., Providence, R.I., 2014. https://doi.org/10.1090/memo/1094
Ruelle, D., Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.
Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. https://doi.org/10.1090/S0002-9904-1967-11798-1
Thomsen, K., $C^*$-algebras of homoclinic and heteroclinic structure in expansive dynamics, vol. 206, Mem. Amer. Math. Soc., no. 970, Amer. Math. Soc., Providence, R.I., 2010. https://doi.org/10.1090/S0065-9266-10-00581-8
Thomsen, K., The homoclinic and heteroclinic $C^*$-algebras of a generalized one-dimensional solenoid, Ergodic Theory Dynam. Systems 30 (2010), no. 1, 263–308. https://doi.org/10.1017/S0143385709000042
Wieler, S., Smale spaces via inverse limits, Ergodic Theory Dynam. Systems 34 (2014), no. 6, 2066–2092. https://doi.org/10.1017/etds.2013.19
Williams, R. F., One-dimensional non-wandering sets, Topology 6 (1967), no. 4, 473–487. https://doi.org/10.1016/0040-9383(67)90005-5
Williams, R. F., Classification of one dimensional attractors, in “Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968)'', Amer. Math. Soc., Providence, R.I., 1970, pp. 341--361.
Yi, I., Canonical symbolic dynamics for one-dimensional generalized solenoids, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3741–3767. https://doi.org/10.1090/S0002-9947-01-02710-6>