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On some generalizations of skew-shifts on $\mathbb{T}^{2}$

Published online by Cambridge University Press:  04 May 2017

KRISTIAN BJERKLÖV
KRISTIAN BJERKLÖV*
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden email bjerklov@kth.se
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Abstract

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, 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 non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. 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 $g$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 1 , January 2019 , pp. 19 - 61
Copyright
© Cambridge University Press, 2017 

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