We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if $X$ is an arc-like continuum which admits a homeomorphism $f$ with positive topological entropy, then $X$ contains an indecomposable subcontinuum. Barge and Diamond proved that if $G$ is a finite graph and $f:G \rightarrow G$ is any map with positive topological entropy, then the inverse limit space $\varprojlim(G,f)$ contains an indecomposable continuum. In this paper we show that if $X$ is a $G$-like continuum for some finite graph $G$ and $f:X \rightarrow X$ is any map with positive topological entropy, then $\varprojlim (X,f)$ contains an indecomposable continuum. As a corollary, we obtain that in the case that $f$ is a homeomorphism, $X$ contains an indecomposable continuum. Moreover, if $f$ has uniformly positive upper entropy, then $X$ is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.