Sequence entropy of rank one systems

We study the sequence entropy of rank one measure-preserving systems along subexponential sequences. We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein’s probabilistic constructions. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.

💡 Research Summary

The paper investigates the sequence entropy h_A(T) of measure‑preserving dynamical systems (X, B, μ, T) when the observation times are taken along a prescribed increasing integer sequence A={t_n}. The focus is on rank‑one systems, i.e., systems that can be built by an iterated cut‑and‑stack construction with a single Rokhlin tower at each stage. The authors first recall the definition of sequence entropy as the asymptotic average Shannon information of the A‑names of a finite partition, and they introduce the symbolic coding associated with the natural tower partitions ξ_n.

The main positive result (Theorem 1.2) states that for any sequence A that “dilates to infinity’’—meaning the gaps t_{n+1}−t_n tend to infinity—one can construct a rank‑one system with infinite sequence entropy, h_A(T)=+∞. The construction is probabilistic: at each stacking step the presence or absence of a spacer level is chosen independently as a Bernoulli trial. By interpreting the spacer pattern as a random word, the authors prove a key lemma (Proposition 3.2) showing that, with high probability, the A‑names of points are sufficiently diverse so that the average information does not vanish. This argument answers a question raised in