1,720,984 research outputs found
Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow
No full textBy a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Moller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: Z(s) = det(1 - L-s(+)) det(1 - L-s(-)). In this article we show that the operator families L-s(+/-) arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for s is an element of C, Res = 1/2, the operator L-s(+) (respectively L-s(-)) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue s(1 - s). For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips-Sarnak conjecture on non-existence of even Maass cusp forms.Volkswagen Foundatio
A DYNAMICAL APPROACH TO MAASS CUSP FORMS
No full textFor nonuniform cofinite Fuchsian groups Gamma that satisfy a certain additional geometric condition, we show that the Maass cusp forms for Gamma are isomorphic to 1-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.ERC Starting Grant ANTHO
A Thermodynamic Formalism Approach to the Selberg Zeta Function for Hecke Triangle Surfaces of Infinite Area
No full textWe provide an explicit construction of a cross section for the geodesic flow on infinite-area Hecke triangle surfaces, which allows us to conduct a transfer operator approach to the Selberg zeta function. Further, we construct closely related cross sections for the billiard flow on the associated triangle surfaces and endow the arising discrete dynamical systems and transfer operator families with two weight functions, which presumably encode Dirichlet respectively Neumann boundary conditions. The Fredholm determinants of these transfer operator families constitute dynamical zeta functions, which provide a factorization of the Selberg zeta function of the Hecke triangle surfaces.Volkswagen Foundatio
Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension
No full textRecently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces Gamma \ G, where G is any connected semisimple Lie group of real rank one with finite center, and 0 is any non-uniform lattice in G. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.EPSRC; SNF [200021-127145]; ERC Starting Grant ANTHO
Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds
No full textWe construct cross sections for the geodesic flow on the orbifolds Gamma\H which. are tailor-made for the requirements of transfer operator approaches to Maass cusp farms and Selberg zeta functions. Here, H denotes the hyperbolic plane and Gamma is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic
The category of reduced orbifolds in local charts
Get PDFIt is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective etale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category
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