LINKED TWIST MAPPINGS HAVE THE K‐PROPERTY

Ergodicity of toral linked twist mappings

“…The case kl>0 has been done by Burton and Easton in [2] and by Wojtkowski in [7]. Almost hyperbolicity when ap< -4 follows from Wojtkowski paper [7].…”

“…Almost hyperbolicity when ap< -4 follows from Wojtkowski paper [7]. Vkl>0, global stable and unstable manifolds intersect each other since they are very long and go, roughly speaking, in different directions.…”

Topological Entropy and Secondary Folding

“…A subtle but important point here is that there is a set of singularities, S ∈ R, where the unstable (and stable) manifolds don't exist [18,28]. This set consists of all points on the boundaries of the R L , R V , and R H regions, as well as all (forward and backward) images of these points.…”

“…(These singular points eventually become the vertices of bends in line segments.) Fortunately, S is a set of measure zero [28], and tangent vectors are well defined in the neighborhood of the singularities. So, using continuity, we will speak of having an 'unstable' manifold at each singular point, even though it may be the vertex of a bend.…”

“…However, we add another restriction and require κλ < −4. This condition ensures that matrix L is hyperbolic and, consequently, that the toral LTM has an ergodic partition [28]. (With a slightly stronger condition, the toral LTM will also be Bernoulli [19].)…”