NP-hardness of deciding f_μ^n(x) < 1/2 for the tent map

Determine whether the decision problem “Given parameter μ ∈ (1,2), real x ∈ [0,1), and integer n, decide if the n-th iterate f_μ^n(x) of the tent map f_μ satisfies f_μ^n(x) < 1/2” is NP-hard under standard polynomial-time reductions.

Background

The paper studies the space complexity of computing tent codes and defines the decision problem of determining whether f_μ^n(x) < 1/2. While the authors develop smoothed-space results, they note that the time complexity and complexity classification of this decision problem remain unsettled. In their discussion of related work, they explicitly state uncertainty about NP-hardness of this particular tent-map decision problem.