What is meant by the statement that a continuous map of an interval $I$ into itself has a horseshoe? State without proof the properties of such a map.

Define the property of chaos of such a map according to Glendinning.

A continuous map $f: I \rightarrow I$ has a periodic orbit of period 5 , in which the elements $x_{j}, j=1, \ldots, 5$ satisfy $x_{j}<x_{j+1}, j=1, \ldots, 4$ and the points are visited in the order $x_{1} \rightarrow x_{3} \rightarrow x_{4} \rightarrow x_{2} \rightarrow x_{5} \rightarrow x_{1}$. Show that the map is chaotic. [The Intermediate Value theorem can be used without proof.]