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Dynamics of composition operators induced by odometers

Udayan B. Darji, Daniel Gomes, Régis Varão

TL;DR

This work analyzes the dynamics of composition operators $T_f$ on $L^p$ spaces induced by non-singular transformations, with a focus on odometer systems in finite measure spaces. It shows a rigidity phenomenon: when $\mu(X)<\infty$, supercyclicity of $T_f$ implies hypercyclicity, and for odometer-induced $T_f$ many chaotic notions collapse so that Li–Yorke chaos, supercyclicity, hypercyclicity, and Devaney chaos become equivalent; yet the Frequent Hypercyclicity Criterion can fail, revealing a sharp distinction from weighted backward shifts. The authors also provide a constructive counterexample: a mixing, chaotic, and distributionally chaotic $T_f$ that does not satisfy the FH-Criterion, highlighting that odometer dynamics deviate substantially from classical shift-like models. Finally, they exhibit an explicit odometer whose associated $T_f$ is mixing and distributionally chaotic while maintaining Devaney chaos and violating FH-Criterion, underscoring both rigidity and separation phenomena in finite-measure linear dynamics. The results illuminate the limitations of translating intuition from dissipative or infinite-measure settings to conservative, finite-measure odometer contexts, with potential implications for operator theory and ergodic dynamics.

Abstract

We study the linear dynamics of composition operators induced by measurable transformations on finite measure spaces, with particular emphasis on operators induced by odometers. Our first main result shows that, on a finite measure space, supercyclicity of a composition operator implies hypercyclicity. This phenomenon has no analogue in several classical settings and highlights a rigidity specific to the finite-measure context. We then focus on composition operators induced by odometers and show that many dynamical properties that are distinct for weighted backward shifts collapse in this setting. In particular, for such operators, supercyclicity, Li-Yorke chaos, hypercyclicity, weak mixing, and Devaney chaos are all equivalent. In contrast to this collapse, we show that the classical equivalence between Devaney chaos and the Frequent Hypercyclicity Criterion for weighted backward shifts fails for odometers. Specifically, we construct a mixing, chaotic, and distributionally chaotic composition operator that does not satisfy the Frequent Hypercyclicity Criterion. This combination of rigidity and separation demonstrates that the dynamical behavior of composition operators induced by odometers differs sharply from that of weighted backward shifts.

Dynamics of composition operators induced by odometers

TL;DR

This work analyzes the dynamics of composition operators on spaces induced by non-singular transformations, with a focus on odometer systems in finite measure spaces. It shows a rigidity phenomenon: when , supercyclicity of implies hypercyclicity, and for odometer-induced many chaotic notions collapse so that Li–Yorke chaos, supercyclicity, hypercyclicity, and Devaney chaos become equivalent; yet the Frequent Hypercyclicity Criterion can fail, revealing a sharp distinction from weighted backward shifts. The authors also provide a constructive counterexample: a mixing, chaotic, and distributionally chaotic that does not satisfy the FH-Criterion, highlighting that odometer dynamics deviate substantially from classical shift-like models. Finally, they exhibit an explicit odometer whose associated is mixing and distributionally chaotic while maintaining Devaney chaos and violating FH-Criterion, underscoring both rigidity and separation phenomena in finite-measure linear dynamics. The results illuminate the limitations of translating intuition from dissipative or infinite-measure settings to conservative, finite-measure odometer contexts, with potential implications for operator theory and ergodic dynamics.

Abstract

We study the linear dynamics of composition operators induced by measurable transformations on finite measure spaces, with particular emphasis on operators induced by odometers. Our first main result shows that, on a finite measure space, supercyclicity of a composition operator implies hypercyclicity. This phenomenon has no analogue in several classical settings and highlights a rigidity specific to the finite-measure context. We then focus on composition operators induced by odometers and show that many dynamical properties that are distinct for weighted backward shifts collapse in this setting. In particular, for such operators, supercyclicity, Li-Yorke chaos, hypercyclicity, weak mixing, and Devaney chaos are all equivalent. In contrast to this collapse, we show that the classical equivalence between Devaney chaos and the Frequent Hypercyclicity Criterion for weighted backward shifts fails for odometers. Specifically, we construct a mixing, chaotic, and distributionally chaotic composition operator that does not satisfy the Frequent Hypercyclicity Criterion. This combination of rigidity and separation demonstrates that the dynamical behavior of composition operators induced by odometers differs sharply from that of weighted backward shifts.
Paper Structure (6 sections, 11 theorems, 71 equations)

This paper contains 6 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Proof of Theorem A
  4. Proof of Theorem B
  5. Proof of Theorem C
  6. Proof of Theorem D

Key Result

Theorem A

Let $(X,\mathcal{B},\mu)$ be a finite measure space and $f:X\to X$ be a measurable non-singular transformation such that $T_f:L^p(X;\mathbb{K})\to L^p(X;\mathbb{K})$ is continuous. If $T_f$ is supercyclic, then it is hypercyclic.

Theorems & Definitions (28)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem 3.1
  • proof
  • Remark
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 18 more