An Exceptional Set in the Ergodic Theory of Markov Maps of the Interval

It is known that a Markov map T of the unit interval preservesa measure µ, say, equivalent to Lebesgue measure, andthat almost every point of the interval has a forward orbitunder T that is uniformly distributed with respect to µ.In the opposite direction the main result of this paper statesthat there is a set of points having Hausdorff dimension 1 whoseforward orbits are in a certain sense very far from being sodistributed. 1991 Mathematics Subject Classification: 58F08,28A80.