Transversality and Lefschetz numbers for foliation maps
Jes\'us A. \'Alvarez L\'opez, Yuri A. Kordyukov

TL;DR
This paper develops a topological approach to defining the Lefschetz number for foliation maps with invariant measures, extending transversality results and exploring connections to analytic Lefschetz numbers and trace formulas.
Contribution
It introduces a topological definition of the $ ext{Lambda}$-Lefschetz number for foliation maps, extending smooth approximation and transversality to foliations, and investigates its relation to analytic Lefschetz numbers.
Findings
Defined topological $ ext{Lambda}$-Lefschetz number for foliation maps.
Extended transversality results to the context of foliations.
Connected the topological and analytic Lefschetz numbers, proposing a Lefschetz trace formula.
Abstract
Let be a smooth foliation on a closed Riemannian manifold , and let be a transverse invariant measure of . Suppose that is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the -Lefschetz number of any leaf preserving diffeomorphism is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological -Lefschetz number is equal to the analytic -Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to .
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Taxonomy
TopicsMathematical Dynamics and Fractals · Geometric Analysis and Curvature Flows
