  In this expository note, we discuss spatially inhomogeneous quantum walks in
one dimension and describe a genre of mathematical methods that enables one to
translate information about the time-independent eigenvalue equation for the
unitary generator into dynamical estimates for the corresponding quantum walk.
To illustrate the general methods, we show how to apply them to a 1D coined
quantum walk whose coins are distributed according to an element of the
Thue--Morse subshift.
