defined; by analogy with the Interval Translation Maps of we surmise that there are cases where a Hausdorff measure restricted to $X^+$ is invariant, in particular in cases where $\ell(X^+)=0$.

Acknowledgements {#acknowledgements .unnumbered}
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The research of PA and XF is supported by EPSRC grant GR/M36335, while K[Ż]{} acknowledges the support by a KBN Grant. PA, TN and K[Ż]{} thank the Max-Planck-Institute for the Physics of Complex Systems, Dresden for providing an opportunity to meet and discuss this paper. We also thank the anonymous referees for their invaluable comments, in particular to one who pointed out a considerable simplification of our argument in Section 2.

Appendix: Proof of Proposition \[propsimplecases\] {#secexample3 .unnumbered}
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We consider the case $-1\leq \alpha<0$ and $0<A<1$; figure \[fig\_paraconst\] shows how the linear map $M_{A, \alpha}$ maps the square $[0,1]^2$ given by $OPRQ$ into the maximal invariant set consisting of the union of the line $OS$ and the triangles $OAB$ and $CSQ$, where $$A=(0,-\alpha),~~B=(A,-\alpha A),~~C=(1-A,-\alpha(1-A))~~\mbox{ and }~~
S=(1.-\alpha).$$ To see this is the maximal invariant set, note that everything in $OPRS$ decreases in its $y$- component unless it lands in the triangle $OAB$. Similarly, all points in $OQS$ must increase in $y$-component unless it lands in 