Quotients of dynamical systems and chaos on the Cantor fan

Let $(X,f)$ be a dynamical system. Using an equivalence relation $\sim$ on $X$, we introduce the quotient $(X{\mathrm{/}}_{\sim },f^{\star })$ of the dynamical system $(X,f)$. In the first part of the paper, we give new results about sensitive dependence on initial conditions of $(X{\mathrm{/}}_{\sim },f^{\star })$, transitivity of $(X{\mathrm{/}}_{\sim },f^{\star })$, and periodic points in $(X{\mathrm{/}}_{\sim },f^{\star })$. In the second part of the paper, we use these results to study chaotic functions on the Cantor fan. Explicitly, we study functions $f$ on the Cantor fan $C$ such that (1) $(C,f)$ is chaotic in the sense of Devaney, (2) $(C,f)$ is chaotic in the sense of Robinson but not in the sense of Devaney, and (3) $(C,f)$ is chaotic in the sense of Knutzen but not in the sense of Devaney. We also study chaos on the Lelek fan.