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Keplerian shear for Chacon Transformations

Arthur Boos, Benoit Saussol

TL;DR

The paper constructs a random-Chacon rank-one family with spacers drawn from $\{1,2\}$ and shows that the resulting fibred system exhibits keplerian shear for odd $d\ge 7$. Central to the argument is a time-dependent local limit theorem for non-i.i.d. Birkhoff sums along the full shift, proven via a complex-perturbation Perron–Frobenius theory on a cone of bounded oscillation functions. The analysis carefully controls exceptional sets where mixing may fail and establishes linear variance growth, spectral data, and moderate deviations to obtain sharp LLT error terms. These ingredients are then combined to deduce conditional strong mixing on invariant fibers and, through Dynkin-class arguments, convergence of autocorrelations on cylinders and then on the entire space. The work thus demonstrates keplerian shear for a non-ergodic, weakly mixing Chacon-type transformation, with quantitative control mediated by the time-dependent LLT and exceptional-set analysis.

Abstract

The concept of keplerian shear was introduced by Damien Thomine recently. It is useful for non ergodic systems, and can be seen as strong mixing conditionally on invariant fibers. The notion is particularly interesting when a.e. fiber is not strongly mixing. We develop here an approach appropriate for systems such that a.e. fiber is weakly mixing, and apply it to a family of rank one transformations. Each transformation is a kind of Chacon map, built with a random number of spacers at each step of the Rochklin tower. We prove that this new dynamical system exhibits keplerian shear. The method relies on a version of a local limit theorem for time dependent Birkhoff sums along the fullshift.

Keplerian shear for Chacon Transformations

TL;DR

The paper constructs a random-Chacon rank-one family with spacers drawn from and shows that the resulting fibred system exhibits keplerian shear for odd . Central to the argument is a time-dependent local limit theorem for non-i.i.d. Birkhoff sums along the full shift, proven via a complex-perturbation Perron–Frobenius theory on a cone of bounded oscillation functions. The analysis carefully controls exceptional sets where mixing may fail and establishes linear variance growth, spectral data, and moderate deviations to obtain sharp LLT error terms. These ingredients are then combined to deduce conditional strong mixing on invariant fibers and, through Dynkin-class arguments, convergence of autocorrelations on cylinders and then on the entire space. The work thus demonstrates keplerian shear for a non-ergodic, weakly mixing Chacon-type transformation, with quantitative control mediated by the time-dependent LLT and exceptional-set analysis.

Abstract

The concept of keplerian shear was introduced by Damien Thomine recently. It is useful for non ergodic systems, and can be seen as strong mixing conditionally on invariant fibers. The notion is particularly interesting when a.e. fiber is not strongly mixing. We develop here an approach appropriate for systems such that a.e. fiber is weakly mixing, and apply it to a family of rank one transformations. Each transformation is a kind of Chacon map, built with a random number of spacers at each step of the Rochklin tower. We prove that this new dynamical system exhibits keplerian shear. The method relies on a version of a local limit theorem for time dependent Birkhoff sums along the fullshift.
Paper Structure (18 sections, 63 theorems, 363 equations, 3 figures)

This paper contains 18 sections, 63 theorems, 363 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Chacon transformation
  3. Returns and induction
  4. Distribution of $\ell-$th returns
  5. Exceptional sets
  6. First exceptional set
  7. Second exceptional set
  8. Limit theorems for time dependent observables
  9. Variance of the non homogeneous Birkhoff sum
  10. Complex perturbations of Perron-Frobenius operator
  11. Moderate deviations
  12. Local limit theorem
  13. Keplerian shear for Chacon transformation
  14. Equivalence of variances
  15. Final estimate of self correlation
  16. ...and 3 more sections

Key Result

Proposition 1.1

Eins$T$ is weak mixing if and only if for all $A,B\in\mathcal{T}$, there exists $J_{A,B}\subset \mathbb{N}$ with density zero such that

Figures (3)

  • Figure 1: First step of the Rockhlin tower for the $d=3$-Chacon transformation
  • Figure 2: Second step of the Rockhlin tower for $d=3$ and $\alpha_0=2$.
  • Figure 3: Second step of the Rockhlin tower for the $d=3$-Chacon Transformation

Theorems & Definitions (133)

  • Proposition 1.1
  • Proposition 1.2
  • Definition 2.1: Chacon transformation
  • Theorem 2.1: Keplerian shear of Chacon transformation
  • Definition 3.1: First return
  • Definition 3.2: Induction
  • Definition 3.3
  • Lemma 3.1: Index property
  • proof
  • Lemma 3.2
  • ...and 123 more