Given a dynamical system $(X,f)$, an orbit is a sequence $(y_i)_{i>=0}$ such that $y_i=f(y_{i-1})$ for all $i>= 1$. Given $delta > 0$, a $delta$-pseudo-orbit is a sequence of points $(x_i)_{i>=0}$ such that $d(x_i, f(x_{i-1}))< delta$ for all $i>=1$. We say that $(X,f)$ has the shadowing property if for every $epsilon>0$, there exists $delta > 0$ such that every $delta$-pseudo-orbit $(x_i)_{i>=0}$ corresponds to a real orbit $(y_i)_{i>=0}$ with $d(x_i,y_i) < epsilon$ for all $i>= 0$.
We investigate shadowing and other dynamical properties in countable state shifts over a compact domain. We demonstrate that when the underlying metric on the countable alphabet is not the discrete metric, then it is possible for a shift of finite order to fail to have the shadowing property; this contrasts with the fact that all shifts of finite order over the product of the discrete metric have shadowing. We provide sufficient conditions for our shifts of order two to have the shadowing property.
This is joint work with James P. Kelly.
