The invariant Cantor set of Smale's horseshoe map is imbedded into a connented and not arcwise connected, invariant, compact set with zero Lebesgue measure.
"Connected and not arcwise connected invariant sets for some 2-dimmensional dynamical systems." J. Math. Kyoto Univ. 49 (2) 339 - 346, 2009. https://doi.org/10.1215/kjm/1256219160