Spectral theory for transfer operators on compact quotients of Euclidean buildings

In this paper we generalize the geodesic flow on (finite) homogeneous graphs to a multiparameter flow on compact quotients of Euclidean buildings. Then we study the joint spectra of the associated transfer operators acting on suitable Lipschitz spaces. The main result says that outside an arbitrarily small neighborhood of zero in the set of spectral parameters the Taylor spectrum of the commuting family of transfer operators is contained in the joint point spectrum.

💡 Research Summary

The paper develops a spectral theory for a family of commuting transfer operators associated with a multiparameter dynamical system on compact quotients of Euclidean buildings. Starting from the well‑known geodesic flow on finite homogeneous graphs, the authors replace the one‑dimensional shift by an action of the dominant coweight monoid (P^{\vee}{+}) on the space of sectors (S(C)) of a compact local building (C). Each dominant coweight (\mu) defines a shift map (\sigma{\mu}) that translates a sector by (\mu) inside the fundamental sector of the underlying Coxeter complex.

To analyse these dynamics the authors endow (S(C)) with a family of ultrametrics (d_{\vartheta}) (with (0<\vartheta<1)), turning it into a Cantor‑like compact metric space. The corresponding Banach spaces (F_{\vartheta}) of complex‑valued Lipschitz functions serve as the functional setting for the transfer operators \