2016/10/11 by Kristian Bjerklöv, Bjerklöv, Kristian
Mathematics · #37C40 #37C70 #37E30 #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1610.03213
openalex publication_date 2016/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we investigate maps of the two-torus \mathbbT2 of the form T(x,y)=(x+ω,g(x)+f(y)) for Diophantine ω∈\mathbbT and for a class of maps f,g:\mathbbT→\mathbbT, where each g is strictly monotone and of degree 2, and each f is an orientation preserving circle homeomorphism. For our class of f and g we show that T is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two T-invariant graphs. One of the graphs is a Strange Nonchaotic Attractor whose basin of attraction consists of (Lebesgue) almost all points in \mathbbT2. Only a low regularity assumption (Lipschitz) is needed on the maps f and g, and the results are robust with respect to Lipschitz-small perturbations of f and g.