Topological entropy and IE-tuples of indecomposable continua
For a graph $G$ we define a new notion of “free tracing property by free $G$-chains” on $G$-like continua and we prove that a positive topological entropy homeomorphism $f$ of a $G$-like continuum $X$ admits a Cantor set $Z$ in $X$ and an indecomposable subcontinuum $H$ of $X$ satisfying the following conditions:
(1) $Z$ has the free tracing property by free $G$-chains,
(2) $H$ is the unique minimal subcontinuum of $X$ containing $Z$ and no two points of $Z$ belong to the same composant of $H$,
(3) any sequence $(z_1,\ldots,z_n)$ of points in $Z$ is an IE-tuple of $f$, and
(4) $f$ is Li–Yorke chaotic on $Z$.