A dynamical lefschetz trace formula for algebraic anosov diffeomorphisms

Abstract

Dynamical Lefschetz trace formulas for flows on compact manifolds were first conjectured by GUILLEMIN [5] and later but independently by PATTERSON [12]. These formulas express the alternating sum of the traces of the induced flow on cohomology by a sum over contributions from the compact orbits. Here the cohomology theory depends on the choice of a foliation which is respected by the flow. This note makes a contribution to PATTERSON's setup. He looks at Anosov flows and considers the unstable foliation. An interesting example of this situation is provided by the geodesic flow on the sphere bundle of a cocompact quotient of PSLe (]~) by a lattice. Using representation theory, PATTERSON was able to define a trace on the infinite dimensional foliation cohomologies. The dynamical Lefschetz trace formula was then shown to be a consequence of the Selberg trace formula. Incidentally this example also fits into GUILLEMIN'S framework who treated it in [5]. Much more work in the context of more general symmetric spaces was done by BUNKE, DEITMAR, JUHL, OLBRICH and SCHUBERT. We re fe r the r eade r to the book of JUHL [7] for a comprehensive overview. Among all Anosov flows there are the ones with an integrable complement or equivalently, those where the sum of the stable and the unstable bundle is integrable. One gets examples by suspending Anosov diffeomorphisms and conjecturally all Anosov flows with an integrable complement should arise in this way up to a time change by a constant factor, [13] w 3. Granting this conjecture the dynamical Lefschetz trace formula for such Anosov flows can be reduced to the following problem: