Interval maps with return index zero (Q6594827)

Piecewise \(C^2\) maps of the interval with indifferent fixed points provide a natural setting for the dynamics of infinite measure-preserving systems. Here a family of examples of this sort are found for which the infinite invariant measure equivalent to Lebesgue measure is `strongly' infinite, meaning that it is pointwise dual ergodic and has slowly varying return sequences. The former is a property of a conservative ergodic infinite measure-preserving transformation giving a form of pointwise ergodic theorem for a suitable unique sequence of weights (the return sequence of the transformation), and the latter describes asymptotic growth properties of this return sequence related to the degree of infiniteness of the invariant measure; the eponymous return index is a measure of this. Systems with index \(1\) are only mildly infinite and in some sense as close as possible to finite systems, and systems with index \(0\) are strongly infinite and in some sense close to dissipative ones. Here a family of interval maps are constructed that preserve a strongly infinite measure in this sense.