SPECIALITY: Pure Mathematics DIRECTION: Differentiable Dynamical Systems

Let M be a compact manifold and f a self-diffeomorphism of M. For a hyperbolic at-tractor Λ of f, Williams systematically reduced the attractor (Λ, f |Λ) to the inverse limit system (Σg, σg) for some Anosov endomorphism g on a branched manifold K. Furthermore, for expanding attractors, he even reduced topological conjugacy of attractors to shift equiva-lence of the corresponding maps g’s. Anosov endomorphisms represent models of hyperbolic attractors in this sense. However, we show that the later reduction does not hold for general hyperbolic attractors. In this dissertation, we will prove the following Theorem 1 Let a be a hyperbolic toral endomorphism. If f is sufficiently C1 close to a, then f is shift equivalent to a if and only if f is topologically equivalent to a. As a corollary, we have Theorem 2 For general Anosov endomorphisms, inverse limit equivalence cannot imply shift equivalence. Recently, Lan Wen proved that an Anosov endomorphism on a branched surface wit