  Our initial data is a transfer operator $L$ for a continuous,
countable-to-one map $\varphi:\Delta \to X$ defined on an open subset of a
locally compact Hausdorff space $X$. Then $L$ may be identified with a
`potential', i.e. a map $\varrho:\Delta\to X$ that need not be continuous
unless $\varphi$ is a local homeomorphism. We define the crossed product
$C_0(X)\rtimes L$ as a universal $C^*$-algebra with explicit generators and
relations, and give an explicit faithful representation of $C_0(X)\rtimes L$
under which it is generated by weighted composition operators. We explain its
relationship with Exel-Royer's crossed products, quiver $C^*$-algebras of Muhly
and Tomforde, $C^*$-algebras associated to complex or self-similar dynamics by
Kajiwara and Watatani, and groupoid $C^*$-algebras associated to
Deaconu-Renault groupoids.
  We describe spectra of core subalgebras of $C_0(X)\rtimes L$ and use it to
characterise simplicity of $C_0(X)\rtimes L$ and prove the uniqueness theorem
for $C_0(X)\rtimes L$. We give efficient criteria for $C_0(X)\rtimes L$ to be a
Kirchberg algebra, and we discuss relationship between KMS states on the core
subalgebra of $C_0(X)\rtimes L$ and conformal measures for $\varphi$.

Operator Algebras;Dynamical Systems