# Itineraries for Inverse Limits of Tent Maps: a Backward View

## TL;DR

This paper introduces new necessary and sufficient backward orbit conditions for itineraries in inverse limits of tent maps, providing a more natural framework that reveals mode locking phenomena related to kneading sequences.

## Contribution

It offers a novel backward orbit characterization of admissibility, contrasting with previous forward-based conditions, and uncovers mode locking in tent map inverse limits.

## Key findings

- Backward admissibility conditions are not symmetric to forward ones.
- Maximum backward itineraries can lock on intervals of kneading sequences.
- Provides a more natural description of inverse limit dynamics.

## Abstract

Previously published admissibility conditions for an element of $\{0,1\}^{\mathbb{Z}}$ to be the itinerary of a point of the inverse limit of a tent map are expressed in terms of forward orbits. We give necessary and sufficient conditions in terms of backward orbits, which is more natural for inverse limits. These backward admissibility conditions are not symmetric versions of the forward ones: in particular, the maximum backward itinerary which can be realised by a tent map mode locks on intervals of kneading sequences.