 (Chaos\(_{te})\)   A discrete map is chaotic just in case it exhibits topological entropy: Let \(f\) be a discrete map and \(\{W_{i}\}\) be a partition of a bounded region \(W\) containing a probability measure which is invariant under \(f\). Then the topological entropy of \(f\) is defined as  \(h_{T}(f) = \sup_{\{W_i\}h(f,\{W_i\})}\),  where sup is the supremum of the set \(\{W_{i}\}\). 