Rotation Sets for Maps of Tori

Stability of the rotation set of area-preserving toral homeomorphisms

“…Combined with Equation (1), this implies that φ dµ ∞ = v, so the mean rotation vector of µ ∞ is v and therefore v ∈ ρ(f ), since the set of mean rotation numbers of invariant probabilities coincides with ρ(f ) (see [MZ89]). …”

“…Note that k j=0 (i j −i j−1 −1) = n−k; thus u = 1 n−k k j=0 (i j −i j−1 −1)v j is a convex combination of the vectors v j , implying that u ∈ ρ(f ). Moreover, Note also that since u ∈ ρ(f ), there exists an invariant measure µ for f with mean rotation vector φdµ = u, where φ : T 2 → R 2 is the map induced byf − id (see [MZ89]). This implies that u lies in the convex hull of φ(T 2 ).…”

“…About (2), it is known that the rotation set is compact and convex [MZ89], every convex polygon of rational vertices is the rotation set of some homeomorphism [Kwa92], but there are non-polygonal examples [Kwa95,BdCH16] (although the known ones are "almost polygonal": they have countably many extremal points).…”

“…It is a compact set defined (in [MZ89]) as all the possible limits of the form lim n→∞ (f n k (z k ) − z k )/n k where (z k ) k∈N is a sequence of points of R 2 and n k → ∞. Any other lift of f has the same rotation set up to an integer translation, so we may refer to the rotation set ρ(f ) of f , regarding it as the set ρ(f ) modulo integer translations 1 .…”

Functions for relative maximization

“…For rotation sets arising from n-dimensional torus continuous maps homotopic to the identity, a result in the spirit of the previous proposition was demonstrated by Misiurewicz and Ziemian (see the theorem 2.10 of [25]) and by Herman (see section 10 of the chapter 1 of [10]). …”