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Coexistence of mixing and rigid behaviors in ergodic theory

Rigoberto Zelada

TL;DR

The paper develops the theory of rigidity groups for adequate sequences and shows that several algebraic, spectral, and unitary characterizations are equivalent, providing a unifying framework for rigidity phenomena. By linking H(U,(n_k)) to a subgroup $G\subseteq\mathbb{Z}^\ell$ and employing Bochner-type limits with measures on the torus, the authors derive interpolation results that apply to generic Lebesgue-preserving automorphisms and to IP-ergodic theory. These results yield new ergodic Ramsey-type examples, including IP$^*$-recurrence obstructions for polynomial families and extensions to broad classes of sequence families, beyond polynomials. The work connects rigidity, weak mixing, Gaussian systems, and spectral theory to combinatorial structures, offering tools to construct transformations with prescribed limiting behaviours and to understand the boundaries of IP$^*$-recurrence in multi-dimensional settings, with potential implications for additive combinatorics and ergodic Ramsey theory.

Abstract

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, there is no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some $|a_j|=1$ if and only if there is an increasing sequence of natural numbers $(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq \{1,...,\ell\}$ there is a $μ$-preserving transformation $T_F:[0,1]\rightarrow[0,1]$ ($μ$ denotes the Lebesgue measure) such that for any measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}μ(A\cap T_F^{-b_jn_k}B)=\begin{cases} μ(A\cap B),\,\text{ if }j\in F,\\ μ(A)μ(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this result has a natural extension to a wide class of families of sequences.

Coexistence of mixing and rigid behaviors in ergodic theory

TL;DR

The paper develops the theory of rigidity groups for adequate sequences and shows that several algebraic, spectral, and unitary characterizations are equivalent, providing a unifying framework for rigidity phenomena. By linking H(U,(n_k)) to a subgroup and employing Bochner-type limits with measures on the torus, the authors derive interpolation results that apply to generic Lebesgue-preserving automorphisms and to IP-ergodic theory. These results yield new ergodic Ramsey-type examples, including IP-recurrence obstructions for polynomial families and extensions to broad classes of sequence families, beyond polynomials. The work connects rigidity, weak mixing, Gaussian systems, and spectral theory to combinatorial structures, offering tools to construct transformations with prescribed limiting behaviours and to understand the boundaries of IP-recurrence in multi-dimensional settings, with potential implications for additive combinatorics and ergodic Ramsey theory.

Abstract

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of , IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given , there is no orthogonal vector with some if and only if there is an increasing sequence of natural numbers with the property that for each there is a -preserving transformation ( denotes the Lebesgue measure) such that for any measurable , We remark that this result has a natural extension to a wide class of families of sequences.

Paper Structure

This paper contains 29 sections, 25 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Historical motivation and background
  3. Definition and characterizations of a rigidity group
  4. Applications of \ref{['0.thm:MainResult']}
  5. A consequence of (ii)$\iff$(iii')
  6. An example in ergodic-Ramsey theory derived from (ii)$\implies$(iii)
  7. Further examples in ergodic-Ramsey Theory
  8. A special case of (ii)$\implies$(iii) in \ref{['0.thm:MainResult']}
  9. The proof of \ref{['1.PropBochnerObservation']}
  10. Step 1: Defining $(n_k)_{k\in\mathbb{N}}$ and the sequences $(\alpha^{(j)}_k)_{k\in\mathbb{N}}$, $j\in\{1,...,\ell\}$
  11. Step 2: Defining $\Psi$ and $\sigma$.
  12. Step 3: Equation \ref{['1.eq:IndependentDistribution']} holds.
  13. The proof of our main technical result
  14. A generalization of \ref{['0.thm:BKLConsequence']}
  15. The proof of \ref{['4.Lem:UnitaryResult']}
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $\ell\in\mathbb{N}$, let $\phi_1,...,\phi_\ell\in\mathbb{Z}[x]$ be non-constant polynomials with zero constant term, and let $F\subseteq \{1,...,\ell\}$. Set $d=\max_{1\leq j\leq \ell}\deg(\phi_j)$. The following statements are equivalent:

Theorems & Definitions (41)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Cf. Theorem 3.5 in BKLUltrafilterPoly.
  • Corollary 1.9
  • Corollary 1.10
  • Proposition 1.11
  • Theorem 1.13: Theorem 2.1 in BHKNilSystems2005
  • ...and 31 more