Introduction to Hyperbolic Dynamics and Ergodic Theory

Self‐adjoint Laplacians and symmetric diffusions on hyperbolic attractors

“…In Section 3 we first collect well‐known facts on partially and uniformly hyperbolic attractors and Gibbs $$u$$‐measures (and SRB) and then observe that the results from Section 2 apply and yield natural self‐adjoint Laplacians and symmetric diffusions on these attractors (Theorem 3.3). In Section 4 we discuss the case of geodesic flows on negatively curved manifolds [9, 15, 24, 26, 37, 44, 53, 72]. In this special case our results recover a well‐known construction of leafwise Laplacians and diffusions, see for instance [90, 91], and they may be viewed as an extension of this construction to the attractors of dissipative systems.…”

Stability in mechanics and dynamics; a brief historical tour from Newton to modern tools, including Riemannian structures and entropy

“…The Shadowing Theorem can be used to easily prove the Shadowing Lemma. See e.g., [15] for proofs and further details of these theorems; see also [14].…”

Anatole Katok

“…His lectures from a 1971 summer school in Katsiveli on the Black Sea [155] appeared with others on hyperbolic dynamics [1], and together, these became the main early Moscow text on differentiable dynamics. Katok's lectures are a systematic presentation of hyperbolic dynamics based on both studying some of the Western work and on filling the Anosov blueprint [8] for studying the topological dynamics of hyperbolic sets using shadowing ‡ (he did so again in the supplement to [258], which he had also translated § and annotated, and the approach can also be found in [102,127,300]). ¶.…”