  The mode-locking regions of a dynamical system are the subsets of the
parameter space of the system within which there exists an attracting periodic
solution. For piecewise-linear continuous maps, these regions have a curious
chain structure with points of zero width called shrinking points. In this
paper we perform a local analysis about an arbitrary shrinking point. This is
achieved by studying the symbolic itineraries of periodic solutions in nearby
mode-locking regions and performing an asymptotic analysis on one-dimensional
slow manifolds in order to build a comprehensive theoretical framework for the
local dynamics. We obtain leading-order quantitative descriptions for the shape
of nearby mode-locking regions, the location of nearby shrinking points, and
the key properties of these shrinking points. We apply the results to the
three-dimensional border-collision normal form, nonsmooth Neimark-Sacker-like
bifurcations, and grazing-sliding bifurcations in a model of a dry friction
oscillator.
