LS category on laminations with transverse invariant measure

Details LS category on laminations with transverse invariant measure LS category on laminations with transverse invariant measure

2011-12-21

arxiv.org/abs/1112.5009v1

Mathematics

Dynamical Systems

37 pages

Scientific paper

A version of the tangential LS category is introduced for topological laminations with a transverse invariant measure. It is similar to the version given for measurable laminations by the same author, but considering a slightly different type of leafwise contractions: now the leaf wise contractions are continuous on the ambient space. This new measured category is invariant by leafwise homotopy equivalences preserving the transverse measures, and the condition of being zero or positive is a transverse invariant. Unfortunately, the measured category is zero in many examples. However, in that case, the rate of convergence to zero of the quantity involved in its definition gives a new invariant, called secondary measured category. Some properties of the classical LS category are extended to our primary and secondary measured categories, like the cup length lower bound, or the upper bounds given by the dimension or the number of critical points (now the measure of the set of leafwise critical points). The usual tangential LS category is also bounded by the number of certain critical sets. It is also proved that the measured category is semicontinuous when the foliated structure and the transverse invariant measure varies on a fixed manifold, which is a version of a result of W. Singhoff and E. Vogt for the tangential category. It is shown also an interesting relation between the secondary category and the growth of pseudo groups. It is expected further applications to the general problem of the existence of closed geodesics on compact manifolds as we shown in the final section.

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