Dynamical properties of 2-torus parabolic maps (Q842253)

The authors consider a parabolic map of the torus \(f:[0,1)^2 \rightarrow [0,1)^2\) where \(f(x,y)=(x',y')\) is defined by \[ \begin{cases} x'=ax+by \pmod1\\ y'=cx+dy \pmod1. \end{cases} \] Let \(X^+=\bigcap_{n=0}^\infty f^n(x)\). First it is shown that \(X^+\) is invariant and almost closed and that the Lebesgue measure restricted to \(X^+\) is invariant. For some examples, the Lebesgue measure of \(X^+\) is computed and estimated [see also \textit{P. Ashwin, X.C. Fu, T. Nishikawa}, and \textit{K. Zyczkowski}, Nonlinearity 13, No. 3, 819--835 (2000; Zbl 0952.37007)]. When \(f\) is a non-horocyclic map and \(f^{-1}\) exists, it is shown that \(f^{-1}\) is not parabolic. Furthermore, the conjugation of invertible maps with integral coefficients is studied. The authors show that this class of maps can be reduced to a family of one-parameter rigid rotations.