Spectral theory for transfer operators on non-standard symbolic spaces arising from transcendental dynamics

We study the shift map on a non-compact symbolic space $X$ endowed with a metric $\rho$ that is not compatible with the usual product topology. While abstract, this setting is motivated by symbolic codings arising from transcendental entire maps with Cantor bouquet Julia sets, where the natural metric differs from the standard product topology. We study a spectral theory of Ionescu-Tulcea and Marinescu type for transfer operators acting on Banach spaces of locally Hölder functions adapted to this non-standard metric. We prove quasi-compactness of the transfer operator and establish the existence of a spectral gap, showing that the spectral radius is isolated and corresponds to a simple leading eigenvalue equal to the topological pressure. We identify the associated eigenfunction, conformal measure, and Gibbs state and establish a variational principle. The arguments rely on general symbolic assumptions and do not rely on the product topology. As an application, we consider symbolic models arising from hyperbolic transcendental entire maps of finite order.