Effective mapping class group dynamics III: Counting filling closed curves on surfaces
Abstract
We prove a quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length $\leq L$ on an arbitrary closed, orientable, negatively curved surface. More generally, we prove estimates of the same kind for the number of free homotopy classes of filling closed curves of a given topological type on a closed, orientable surface whose geometric intersection number with respect to a given filling geodesic current is $\leq L$. The proofs rely on recent progress made on the study of the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface, and introduce a novel method for addressing counting problems of mapping class group orbits that naturally yields power saving error terms. This method is also applied to study counting problems of mapping class group orbits of Teichmüller space with respect to Thurston's asymmetric metric.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.11386
 Bibcode:
 2021arXiv210611386A
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Geometric Topology
 EPrint:
 34 pages, 6 figures. arXiv admin note: text overlap with arXiv:2104.01694