Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.
Studia Mathematica
MSC: Primary 37A05; Secondary 37A35, 37E05
DOI: 10.4064/sm250313-9-2
Opublikowany online: 25 June 2026
Streszczenie
We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \mathrm{id}$ all such maps have topological entropy at least $\log 2/2$ and generically they have infinite topological entropy. We show that the generic map with respect to the uniform topology has zero measure-theoretic entropy with respect to the Lebesgue measure. This implies that there are dramatic differences in the topological versus measure-theoretic behavior both for injectivity and for the structure of the level sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arc.
