There are many different crossed products by an endomorphism of a C*-algebra,
and constructions by Exel and Stacey have proved particularly useful. Here we
show that every Exel crossed product is isomorphic to a Stacey crossed product,
though by a different endomorphism of a different C*-algebra. We apply this
result to a variety of Exel systems, including those associated to shifts on
the path spaces of directed graphs.

Given an n x n integer matrix A whose eigenvalues are strictly greater than 1
in absolute value, let \sigma_A be the transformation of the n-torus
T^n=R^n/Z^n defined by \sigma_A(e^{2\pi ix})=e^{2\pi iAx} for x\in R^n. We
study the associated crossed-product C*-algebra, which is defined using a
certain transfer operator for \sigma_A, proving it to be simple and purely
infinite and computing its K-theory groups.

Higher-rank graphs were introduced by Kumjian and Pask to provide models for
higher-rank Cuntz-Krieger algebras. In a previous paper, we constructed
2-graphs whose path spaces are rank-two subshifts of finite type, and showed
that this construction yields aperiodic 2-graphs whose $C^*$-algebras are
simple and are not ordinary graph algebras. Here we show that the construction
also gives a family of periodic 2-graphs which we call \emph{domino graphs}.

We consider two categories of C*-algebras; in the first, the isomorphisms are
ordinary isomorphisms, and in the second, the isomorphisms are Morita
equivalences. We show how these two categories, and categories of dynamical
systems based on them, crop up in a variety of C*-algebraic contexts. We show
that Rieffel's construction of a fixed-point algebra for a proper action can be
made into functors defined on these categories, and that his Morita equivalence
then gives a natural isomorphism between these functors and crossed-product
functors.