Markov partitions for hyperbolic toral automorphisms

Thesis (Ph. D.)--University of Washington, 1992The study of the dynamical properties of hyperbolic toral automorphisms is simplified when the automorphisms are represented as shifts of finite type. The conventional method used to represent such an automorphism symbolically is to construct a Markov partition. The existence of Markov partitions for hyperbolic toral automorphisms is known. Because of the fractal nature of the boundary of such a partition an explicit construction is often difficult even in lower dimensions. Adler has suggested that such a construction may be possible by using digit expansions in powers of the automorphism. The basic idea is to generalize the correspondence between $\beta$-expansions, for $\beta$ a Pisot number, and the $\beta$-shift. This is the motivation behind this thesis.We generalize the notion of digit expansions by considering a subset $X \subset \IR\sp{n}$ which is tiled by a periodically self similar tiling ${\cal T}$ with expansion map $\phi$. We show that every point in X has a digit expansion in powers of $\phi$. The sequences of digits correspond to a path in a directed graph $\Gamma$. This is the content of Chapters 1 and 2.In Chapter 3 we consider a hyperbolic automorphism $\phi$ of $\IR\sp{n}$ which is invariant on the integer lattice. We suppose there is a periodically self similar tiling of the unstable eigenspace for $\phi$. By applying the theory of Chapters 1 and 2 we construct digit expansions for the points in the unstable eigenspace in powers of $\phi$. We then identify the unstable eigenspace modulo the integer lattice with points in a compact subset of $\IR\sp{n}$ which we call $\Omega$. We give necessary and sufficient conditions for $\Omega$ to be almost homeomorphic to the n-dimensional torus. We extend the idea of digit expansions to describe each point in $\Omega$ as a bi-infinite series in powers of $\phi$. The corresponding bi-infinite sequence of digits are in one-one correspondence with bi-infinite paths in a directed graph $\Gamma$.In Chapter 4 we assume that $\Omega$ is almost homeomorphic to the n-dimensional torus. We show that the shift of finite type consisting of the bi-infinite paths in $\Gamma$ is metrically similar to the dynamical system induced by $\phi$ on the n-dimensional torus.We conclude in Chapter 5 with examples. In particular we indicate how the $\beta$-shift, for $\beta$ Pisot, may be extended to a shift of finite type which represents a hyperbolic toral automorphism with eigenvalue $\beta$