  In two-parameter bifurcation diagrams of piecewise-linear continuous maps on
$\mathbb{R}^N$, mode-locking regions typically have points of zero width known
as shrinking points. Near any shrinking point, but outside the associated
mode-locking region, a significant proportion of parameter space can be
usefully partitioned into a two-dimensional array of nearly-hyperbolic annular
sectors. The purpose of this paper is to show that in these sectors the
dynamics is well-approximated by a three-parameter family of skew sawtooth
circle maps, where the relationship between the skew sawtooth maps and the
$N$-dimensional map is fixed within each sector. The skew sawtooth maps are
continuous, degree-one, and piecewise-linear, with two different slopes. They
approximate the stable dynamics of the $N$-dimensional map with an error that
goes to zero with the distance from the shrinking point. The results explain
the complicated radial pattern of periodic, quasi-periodic, and chaotic
dynamics that occurs near shrinking points.
