Ergodic Theory and Dynamical Systems



Topological invariants for substitution tilings and their associated $C^\ast$-algebras 1


JARED E. ANDERSON a1 and IAN F. PUTNAM a2
a1 Department of Mathematics, Princeton University, Princeton, NJ 08544–1000, USA
a2 Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3P4, Canada

Abstract

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

(Received December 13 1995)
(Revised August 16 1996)



Footnotes

1 Research supported by NSERC.