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

TL;DR

This paper develops a detailed geometric and quantitative framework for uniformly hyperbolic sets, including stable manifolds, spectral decomposition, shadowing, Markov partitions, and symbolic coding, with explicit bounds and regularity results.

Contribution

It provides explicit quantitative bounds and constructive methods for key hyperbolic dynamics concepts, advancing the geometric theory with precise estimates and regularity controls.

Findings

Proved the Stable Manifold Theorem with explicit regularity and size estimates.

Established the existence of Markov partitions with explicit diameter bounds.

Constructed a Hölder continuous coding map with quantitative control on injectivity.

Abstract

This Part 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 H\"{o}lder dependence of the local stable and unstable manifolds on the base point, with explicit manifold-size estimate in terms of the contraction rate $\lambda$ 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 H\"{o}lder 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 H\"{o}lder 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 Thiam (2026a) and the variational theory of Part II Thiam (2026b) to the smooth setting. This Part constitutes Part III of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.