$\mathcal{C}^{2}$ surface diffeomorphisms have symbolic extensions

Symbolic Dynamics for Surface Diffeomorphisms with Positive Entropy

Part 1. Chains as pseudo–orbits 8 2. Pesin charts 8 2.1. Non-uniform hyperbolicity 8 2.2. Lyapunov change of coordinates 8 2.3. Pesin Charts 10 2.4. Distortion compensating bounds 11 2.5. NUHχ (f) 11 3. Overlapping charts 11 3.1. The overlap condition 12 3.2. The form of f in overlapping charts 14 3.3. Coarse graining 16 4. ε–chains and an infinite-to-one Markov extension of f 18 4.1. Double ch...

Symbolic extensions

Hilbert Space Extensions of Anosov Diffeomorphisms

We consider extensions of Anosov diffeomorphisms on infranilmanifolds with Hilbert space fibers and disprove a conjecture regarding conditions equivalent to transitivity of such extensions.

Transitivity of Euclidean Extensions of Anosov Diffeomorphisms

We consider the class of Rn extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is R, the existence of a semi-orbit with the projection on R unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and s...

Area-preserving Surface Diffeomorphisms

We prove some generic properties for Cr , r = 1, 2, . . . ,∞, areapreserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez [8] on S to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem.