We construct Markov partitions for non-invertible and/or singular nonuniformy hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications. 24 3.3. Double charts 27 3.4. The graph transform method 28 3.5. Stable/unstable manifolds of ε-gpo's 29 3.6. Stable/unstable sets of ε-gpo's 31 3.7. Hyperbolicity along stable/unstable sets 34 4. Coarse graining 37 5. The inverse problem 41

The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer’s theorem for the toy model family.

We construct a version of kneading theory for families of monotonous functions on the real line. The generality of the setup covers two classical results from Milnor-Thurston's kneading theory: the first one is to dynamically characterise an l-modal map by its kneading sequence, the second one is to define the concept of kneading determinant, relate it to topological entropy and use this to construct a certain type of special "linearazing measure".

The goal of this article is to study how combinatorial equivalence implies topological equiconjugacy. For that we introduce kneading sequences for a particular class of nonautonomous discrete dynamical systems and show that these sequences are a complete invariant for topological equiconjugacy classes.