Dense orbits on the interval (Q1109390)

The authors consider continuous maps f: \(I\to I\) of a closed interval into itself. Associated with f is the inverse limit space \((I,f)=\{(x_ 0,x_ 1,...)| \quad x_ i\in I,\quad f(x_{i+1})=x_ i\}\) and the induced homeomorphism \(\hat f:\) (I,f)\(\to (I,f)\) given by \(\hat f(x_ 0,x_ 1,...)=(f(x_ 0),x_ 0,x_ 1,...).\) In the paper the relationship between the dynamics of f on I and of \(\hat f\) on (I,f) is explored. Under the assumption that f 2 has a dense orbit on I conditions for the existence of a dense orbit of \(\hat f\) on (I,f) are derived. Existence and properties of homoclinic and heteroclinic points and other implications are discussed and some examples are given.