NP-hardness of predicting tent map iterates

Determine whether the decision problem for the tent map—given rational parameter μ ∈ (1,2), rational initial value x ∈ [0,1), and integer n encoded in binary—of deciding whether f_μ^n(x) < 1/2 is NP-hard under standard complexity-theoretic reductions.

Background

The paper studies the computational complexity of predicting properties of chaotic sequences generated by the tent map f_μ on [0,1], focusing on the decision problem: given μ, x, and n, decide whether f_μ^n(x) < 1/2. While the work establishes space-efficient algorithms under smoothed analysis for related recognition tasks, it expressly leaves the time complexity of this threshold decision problem unresolved.

The authors conjecture a possible NP-hardness based on analogies to algebraic and number-theoretic complexity, but report they could not locate or prove such a classification. Establishing NP-hardness (or refuting it) would clarify the predictive complexity of chaotic dynamics in a central benchmark case.