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Historical behavior of skew products and arcsine laws

Pablo G. Barrientos, Raul R. Chavez

TL;DR

The paper develops a unified probabilistic mechanism for historical behavior in skew products with one-dimensional fibers by linking weak arcsine fluctuations to non-convergence of Birkhoff averages. It shows that when the fiber process is conjugate to a random walk (or when the base measure is ergodic with a fiber fluctuation law), almost every orbit exhibits historical behavior, and the empirical measures accumulate on convex combinations of endpoint deltas. This framework recovers and extends many known examples (e.g., Bonifant–Milnor, Molinek, Thaler-type maps, skew-flows) and applies to both one-step and mild skew products, including skew-translations and skew-flows, via coboundary/cohomology criteria. The results illuminate a probabilistic pathway to Takens’ last problem by demonstrating how arcsine-type fluctuations can underpin comprehensive non-statistical dynamics in broad skew-product settings with one-dimensional fibers. The work thus provides a cohesive, general method to identify, classify, and extend historical behavior in deterministic systems with random-like fiber dynamics and has implications for understanding physical measures and long-term statistical predictions in non-statistical regimes.

Abstract

We study the occurrence of historical behavior for almost every point in the setting of skew products with one-dimensional fiber dynamics. Under suitable ergodic conditions, we establish that a weak form of the arcsine law leads to the non-convergence of Birkhoff averages along almost every orbit. As an application, we show that this phenomenon occurs for one-step skew product maps over a Bernoulli shift, where the stochastic process induced by the iterates of the fiber maps is conjugate to a random walk. Furthermore, we revisit known examples of skew products that exhibit historical behavior almost everywhere, verifying that they fulfill the required ergodic and probabilistic conditions. Consequently, our results provide a unified and generalized framework that connects such behaviors to the arcsine distribution of the orbits.

Historical behavior of skew products and arcsine laws

TL;DR

The paper develops a unified probabilistic mechanism for historical behavior in skew products with one-dimensional fibers by linking weak arcsine fluctuations to non-convergence of Birkhoff averages. It shows that when the fiber process is conjugate to a random walk (or when the base measure is ergodic with a fiber fluctuation law), almost every orbit exhibits historical behavior, and the empirical measures accumulate on convex combinations of endpoint deltas. This framework recovers and extends many known examples (e.g., Bonifant–Milnor, Molinek, Thaler-type maps, skew-flows) and applies to both one-step and mild skew products, including skew-translations and skew-flows, via coboundary/cohomology criteria. The results illuminate a probabilistic pathway to Takens’ last problem by demonstrating how arcsine-type fluctuations can underpin comprehensive non-statistical dynamics in broad skew-product settings with one-dimensional fibers. The work thus provides a cohesive, general method to identify, classify, and extend historical behavior in deterministic systems with random-like fiber dynamics and has implications for understanding physical measures and long-term statistical predictions in non-statistical regimes.

Abstract

We study the occurrence of historical behavior for almost every point in the setting of skew products with one-dimensional fiber dynamics. Under suitable ergodic conditions, we establish that a weak form of the arcsine law leads to the non-convergence of Birkhoff averages along almost every orbit. As an application, we show that this phenomenon occurs for one-step skew product maps over a Bernoulli shift, where the stochastic process induced by the iterates of the fiber maps is conjugate to a random walk. Furthermore, we revisit known examples of skew products that exhibit historical behavior almost everywhere, verifying that they fulfill the required ergodic and probabilistic conditions. Consequently, our results provide a unified and generalized framework that connects such behaviors to the arcsine distribution of the orbits.
Paper Structure (43 sections, 66 theorems, 256 equations, 3 figures)

This paper contains 43 sections, 66 theorems, 256 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Presentation of results and context
  3. Historical behavior from random walks
  4. Historical behavior from the ergodicity of the reference measure
  5. Limit points of the sequence of empirical measures
  6. Examples
  7. $T^\Psi$-transformations
  8. Coupling random walks
  9. Skew-flow transformations
  10. Zero Schwarzian derivative
  11. Interval functions
  12. Organization of the paper:
  13. Historical behavior on skew product maps
  14. Definition of historical behavior
  15. Historical behavior for extension maps
  16. ...and 28 more sections

Key Result

Theorem A

Let $F$ be a one-step skew product as in one-skew product-principal, satisfying conditions H0--H2 and the pointwise-fiber fluctuation law with constants $\gamma_0<\gamma_1$. Then, for every $x\in (0,1)$, In particular, $F$ exhibits historical behavior for $(\mathbb{P} \times \mathop{\mathrm{Leb}}\nolimits)$-almost every point.

Figures (3)

  • Figure 1: Fiber maps of a coupling random walk skew product with $\mathcal{A} = \{-2, 1, 3\}$ and probability vector $p=(\frac{1}{2}, \frac{1}{4}, \frac{1}{4})$. The one-step random variables are $Z(\omega_0) = \omega_0$, $\Psi(-2) = 2$, $\Psi(1) = -1$, and $\Psi(3) = 1$, using a north-south map $T(u) = u^2$ with partition points $p_k = \frac{2^k}{1 + 2^k}$ for $k \in \mathbb{Z}$.
  • Figure 2: Symmetric Manneville-Pomeau
  • Figure 3: Thaler functions

Theorems & Definitions (143)

  • Definition 1.1
  • Theorem A
  • Definition 1.2
  • Proposition I
  • Definition 1.3
  • Theorem B
  • Proposition II
  • Remark 1.4
  • Corollary III
  • Proposition IV
  • ...and 133 more