automorphism gives a parabolic map whose matrix has bottom row $(0,~*)$. Since conjugation preserves integrality, trace and determinant, the resulting matrix must be of the form $(1,~B;~0,~1)$ for some $B$. Therefore we conclude that all parabolic maps with integer coefficients can be reduced to one of the horocyclic cases.

Maximal invariant sets {#secmaximal}
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By Poincaré recurrence the maximal invariant set $X^+=X$ (up to a set of zero measure) if and only if $f$ is invertible at almost every point. In fact the maximal invariant set $X^+$ is an upper semicontinuous function of the system parameters $A,\alpha$ in the Hausdorff metric. However, the measure (and dimension) of $X^+$ can (and does) change discontinously with parameters; see Proposition \[propsimplecases\].

We now consider the structure of $X^+$ for the semirational case (ie $\alpha$ rational). Consider the family of lines $L_B$ defined by $$y = B- \alpha x$$ with $B$ a fixed real number. projected onto the torus by taking modulo $1$ in $x$ and $y$. This can be thought of as the set $$L_B=\{ (x,y)~:~ y=B+K-\alpha (x+L)~~\mbox{with}~~(K,L)\in{{\bf Z}}^2\}\cap X.$$ Note that if $$y=B+K-\alpha (x+L)$$ with $L$ and $K$ integers such that $x,y$ in $[0,1]$ then $y'=-\alpha A x+ (1-A)(B+K-\alpha (x+L) +N$ and 