Quotients of Dynamical Systems and Chaos on the Cantor Fan

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