Specification property and distributional chaos almost everywhere

Abstract:

Our main result shows that a continuous map $f$ acting on a compact metric space $(X,\rho )$ with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set $S$ dense in $X$ which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points $u,v\in S$, the upper distribution function is identically 1 and the lower distribution function is zero at some $\varepsilon >0$. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when $X$ is the $k$-dimensional cube $I^{k}$. If $X=I$, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.