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Sequence entropy of rank one systems

Shigenori Takeda

TL;DR

This work analyzes the sequence entropy $h^A(T)$ for zero-entropy rank-one systems along subexponential sequences. It shows that $h^A(T)$ can be made infinite for any sequence $A$ dilating to infinity via a probabilistic cut-and-stack construction, and identifies sharp thresholds where subexponential tower growth forces $h^A(T)=0$. It also provides partial flexibility results for polynomial sequences at a critical growth threshold, illustrating a broad spectrum of attainable sequence-entropy values. The results illuminate the interaction between orbit-length sparsity, tower-structure growth, and entropy-type invariants in rank-one dynamics, with implications for the flexibility program in ergodic theory. Overall, the paper advances understanding of when zero-entropy systems can realize large or small sequence entropy along prescribed sequences.

Abstract

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.

Sequence entropy of rank one systems

TL;DR

This work analyzes the sequence entropy for zero-entropy rank-one systems along subexponential sequences. It shows that can be made infinite for any sequence dilating to infinity via a probabilistic cut-and-stack construction, and identifies sharp thresholds where subexponential tower growth forces . It also provides partial flexibility results for polynomial sequences at a critical growth threshold, illustrating a broad spectrum of attainable sequence-entropy values. The results illuminate the interaction between orbit-length sparsity, tower-structure growth, and entropy-type invariants in rank-one dynamics, with implications for the flexibility program in ergodic theory. Overall, the paper advances understanding of when zero-entropy systems can realize large or small sequence entropy along prescribed sequences.

Abstract

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.
Paper Structure (16 sections, 12 theorems, 87 equations, 2 figures)

This paper contains 16 sections, 12 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sequence entropy and symbolic dynamics
  4. Constructive geometric definition of rank one systems:
  5. Refinement of induced partitions
  6. Paper-specific notations
  7. Lower bounds and the probabilistic method
  8. Genericness of high-complexity stacks
  9. Context and outline of proof
  10. Statement and proof
  11. Blow-up of sequence entropy
  12. Cases of zero sequence entropy
  13. Preliminaries
  14. Outline and proof
  15. Establishing the critical threshold
  16. ...and 1 more sections

Key Result

Theorem 1.2

Given any sequence of non-negative integers $A$ that dilates to infinity, there is a rank one system $(X,\mathcal{B},\mu,T)$ such that $h^A(T) = +\infty$.

Figures (2)

  • Figure 1: Construction of random sequences, to be read bottom-up. Each box in $\Phi_r(S_n)$ refers to the word $\Phi_r(S_r)=12 \cdots h_r$. Red lines correspond to spacer symbols $0$. Solid red lines represent the already determined $0^{b_{r,i}}$ levels and may or may not correspond to an existent spacer symbol. Dashed red lines represent the randomised levels $0^{a_i}$.
  • Figure 2: The hierarchy of parameter sizes at each step of the cut-and-stack procedure. Blue hatch marks correspond to the initial symbols of an $A$-orbit that are too predictable, whereas black ones have sufficiently wide gaps in between for adequate mixing.

Theorems & Definitions (28)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • ...and 18 more