On $C^*$-algebras associated to transfer operators for countable-to-one maps

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 $\varphi$.