| For a primitive, aperiodic substitution S with a common prefix, we define (X, T) to be the corresponding substitution dynamical system, and let (Y, Ft) be the Perron-Frobenius suspension of (X, T). This can be viewed as a substitution tiling dynamical system. This dissertation studies the relation between the discrete spectrum Sp(Y) of the flow ( Y, Ft) and the first Čech cohomology Hˇ1(Y, Z ) of the tiling space Y. These are substantial researches in both areas [AP], [BD], [So], [Ro1], [FMN], [CS]. In particular, Hˇ1(Y, Z ) can be identified with the Brushlinski group Br( Y) consisting of circle valued continuous functions up to homotopy. The set Sp(Y) of eigenfunctions consists of continuous circle valued functions up to constant multiples, and the eigenfunctions for different eigenvalues are not homotopic. Thus Sp( Y) is isomorphic to a subgroup of Hˇ 1(Y, Z ).;In Chapter 1, we show that Hˇ1( Y, Z ) is isomorphic to the dynamical cohomology group of the base ( X, T). We prove that if (X, T) is saturated in the sense of Bezuglyi and Kwiatkiwski [BK], then the dynamical cohomology group is isomorphic to the integral group I (X,T), which is the image of the group of integer valued continuous functions under integral operation. With the additional assumption that the substitution matrix is unimodular, I (X,T) is the subgroup of R generated by the measures of single letter cylinder sets.;In Chapter 2, we first prove that a collared substitution forces its borders, and that a border forcing substitution defines a graph map satisfying the flattening axiom. Then we show how to compute Hˇ 1(Y, Z ) as an inverse limit of the graph map.;In Chapter 3, we prove that I (X,T) is equal to the winding number group Wμ(Y) of the suspension dynamical system (Y, Ft). We show that W μ(Y) does not depend of the height function, only on (X, T). Also, we show that if (X, T) is saturated, then I (X,T) is isomorphic to Hˇ 1(Y, Z ). Thus every element of Čech cohomology group has a unique winding number.;Since the winding number of an eigenfunction is its eigenvalue, there is an inclusion of the eigenvalues to the winding numbers. Using this fact, we give a general algorithm for the explicit computation of Sp( Y) for unimodular substitutions with irreducible characteristic polynomials. Varying the height function can give different spectra. We show that for Perron-Frobenius suspensions of the metallic number substitutions the spectrum is isomorphic to the Čech cohomology group. |
