A weighted dynamical version $\mathbf{Z}_f$ of the classical Ihara zeta function is presented on connected finite regular graphs, expressed in terms of weighted periodic orbit data. We establish its meromorphic continuation, with poles given by the resonances of the associated non-backtracking transfer operator acting on a suitable Banach space, and compute its residues. At simple spectral parameters, the residue of $\mathbf{Z}_f$ is identified with the invariant Ruelle distribution on the graph phase space. Combining this result with a relation obtained by Arends-Palmirotta, we further derive Patterson-Sullivan and Wigner residue formulae. This provides a finite-graph analogue of residue formulae for weighted dynamical zeta functions for geodesic flow on rank one locally symmetric spaces, in the spirit of Sch\"utte-Barkhofen-Weich.
We study sums associated with the action of $PGL_2(\mathbb Z)$ on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under su...
For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality betw...
Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of $f$, we prove that the complement of its postcritical divisor is taut unless $f$ is a monomial power map or admits a nontrivial equivariant rationa...
We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta funct...
We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $\Gamma$ for which $$H^1_{\m...
In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that t...
Johannes Droschl· 1 citation
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