Itinerary of an interval map

Given a finite measurable partition of an interval, the itinerary of $x$ records which partition element contains each iterate $T^n(x)$. Prescribing a finite initial word defines an itinerary cylinder. For the full tent map, every length-$N$ binary cylinder has Lebesgue measure $2^{-N}$ up to endpoint conventions, so its itinerary process is a fair i.i.d. Bernoulli process.

An itinerary cylinder is the set of points whose first finitely many itinerary symbols equal a prescribed finite word. Such cylinders generate the symbolic σ-algebra and pull back to finite intersections of inverse images of partition elements.