On chain continuity

A number of recent papers examine for a dynamical system $f: X \rightarrow X$ the concept of equicontinuity at a point. A point $x \in X$ is an equicontinuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that the orbit of initial points $\delta$ close to $x$ remains at all times $\epsilon$ close to the corresponding points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ implies $d(f^i(x),f^i(x_0)) \leq \epsilon$ for $i = 1,2,\ldots$. If we suppose that the errors occur not only at the initial point but at each iterate we obtain not the orbit of $x_0$ but a $\delta$-chain, a sequence $\{x_0,x_1,x_2,\ldots\}$ such that $d(f(x_i),x_{i+1}) \leq \delta$ for $i = 0,1,\ldots$. The point $x$ is called a chain continuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that all $\delta$ chains beginning $\delta$ close to $x$ remain $\epsilon$ close to the points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ and $d(f(x_i),x_{i+1}) \leq \delta$ imply $d(f^i(x),x_i) \leq \epsilon$ for $i = 1,2,\ldots$. In this note we characterize this property of chain continuity. Despite the strength of this property, there is a class of systems $(X,f)$ for which the chain continuity points form a residual subset of the space $X$. For a manifold $X$ this class includes a residual subset of the space of homeomorphisms on $X$.