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arXiv · 2411.02106

Length averages for codimension one foliations

Masayuki Asaoka·Yushi Nakano·Paulo Varandas·Tomoo Yokoyama
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Abstract

In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group structure itself, to obtain irregular behavior of ball averages for certain non-amenable group actions. Several geometric realization results show that any such groups can appear connected with the topology of leaves which are connected sums of plugs with a special geometry, namely nearly equidistant boundary components. This is used to produce the first examples of codimension one $\mathcal C^\infty$ regular foliations on a compact Riemannian manifold $M$ for which the length average of some continuous function does not exist on a non-empty open subset of $M$.

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Masayuki Asaoka, Yushi Nakano, Paulo Varandas, Tomoo Yokoyama. 2024-11-04. Length averages for codimension one foliations. https://arxiv.org/abs/2411.02106

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