Shadowing Property of Continuous Maps

We study continuous maps of an interval into itself. We find the necessary and sufficient condition for the maps of the type 2" to have the shadowing property. Further we show that any chaotic map, which has only cycles of order a power of 2, does not have the shadowing property. Introduction Let /:/(=(0;l))—»7 be a continuous map of the interval 7 to itself (i.e., / G C°(7, 7)). The orbit of x is the set {x, /(x), /V), ... } and let fsix) := (fix) 5; fix) + ô). Given ¿5 > 0, a ¿5-chain of x (or ¿5-pseudo orbit of x) is a sequence X¿ = {x„}^0 , where x,+i G fs(x.¡) and x0 = x . The investigation of the pseudo orbits is very important in connection with the calculation of the orbits by a computer, because a computer can calculate only pseudo orbits. For example, if a computer has accuracy in computation of fix) to 20 decimal places, we can regard x,-+i as the truncation of /(x,) to 20 decimal places so that \xi+x /(x,)| < 2-20 . Therefore it is very interesting to know when the pseudo orbits can be approximated by actual orbits. We say that ¿-chain X¿ is e-shadowed by an orbit of y if for every n