Uniform Hyperbolicity and Symbolic Dynamics: Markov Partitions, Shadowing, and the Coding of Axiom A Systems

Abstract

This paper establishes the geometric theory of uniformly hyperbolic sets with explicit quantitative bounds throughout, and contains five main theorems. The Stable Manifold Theorem is proved via the backward graph transform, with a complete fiber-contraction argument yielding $C^r$ regularity and Hölder dependence of the local stable and unstable manifolds on the base point, with explicit manifold-size estimate in terms of the contraction rate $λ$ and the second-derivative bound of the diffeomorphism. The Spectral Decomposition Theorem gives the unique decomposition of the nonwandering set into basic sets, with explicit mixing rates for the topologically mixing factors. The Shadowing Lemma provides explicit error bounds controlling how far a pseudo-orbit deviates from a tracking true orbit. The existence of Markov partitions of arbitrarily small diameter is established constructively, with explicit diameter bounds expressed in terms of the shadowing constants. Finally, the coding map from the subshift of finite type to the hyperbolic set is constructed and shown to be Hölder continuous with quantitative control on the exceptional set where it fails to be injective. Along the way we establish canonical coordinates through the bracket map with quantitative bounds. All constants are expressed in terms of the contraction rate, the Hölder exponent of the derivative, the manifold dimension, and the injectivity radius, providing the quantitative infrastructure required to transfer the symbolic spectral theory of Part I and the variational theory of Part II to the smooth setting. This paper constitutes Part III of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.