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Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles

Reza Mohammadpour

TL;DR

This work studies entropy in Lyapunov optimization for typical SL(2,R) one‑step cocycles under pinching and twisting. It leverages Barabanov extremal norms and domination on the Mather set to link Lyapunov optimization to additive Birkhoff optimization, and proves that Lyapunov maximizing measures have zero entropy when the invariant directions e1,e2 are one‑to‑one on the Mather set, with explicit examples showing necessity of this injectivity. The paper also situates Lyapunov optimization as a zero‑temperature limit of the subadditive thermodynamic formalism and relates the maximizing measures to a continuous potential on the base. Overall, the results reinforce a low‑complexity, zero‑entropy picture for Lyapunov optimization in non‑commutative dynamics and provide a practical pathway to analyze ground states via almost additive potentials.

Abstract

In this paper we study ergodic optimization problems for typical cocycles. We consider one-step $SL(2,\mathbb{R})$-cocycles that satisfy pinching and twisting conditions. We prove that the Lyapunov maximizing measures have zero entropy under additional assumptions that the maps $e_1$ and $e_2$ are one-to-one on the Mather set.

Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles

TL;DR

This work studies entropy in Lyapunov optimization for typical SL(2,R) one‑step cocycles under pinching and twisting. It leverages Barabanov extremal norms and domination on the Mather set to link Lyapunov optimization to additive Birkhoff optimization, and proves that Lyapunov maximizing measures have zero entropy when the invariant directions e1,e2 are one‑to‑one on the Mather set, with explicit examples showing necessity of this injectivity. The paper also situates Lyapunov optimization as a zero‑temperature limit of the subadditive thermodynamic formalism and relates the maximizing measures to a continuous potential on the base. Overall, the results reinforce a low‑complexity, zero‑entropy picture for Lyapunov optimization in non‑commutative dynamics and provide a practical pathway to analyze ground states via almost additive potentials.

Abstract

In this paper we study ergodic optimization problems for typical cocycles. We consider one-step -cocycles that satisfy pinching and twisting conditions. We prove that the Lyapunov maximizing measures have zero entropy under additional assumptions that the maps and are one-to-one on the Mather set.

Paper Structure

This paper contains 9 sections, 11 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ergodic optimization of Birkhoff averages
  3. Ergodic optimization of Lyapunov exponents
  4. Typical cocycles
  5. Domination and Multicone
  6. Precise setting and statements
  7. Proof of the main theorem
  8. Barabanov functions
  9. Ergodic optimization of Lyapunov exponents of typical cocycles and Birkhoff averages are equivalent

Key Result

Theorem 1.2

Let $\mathcal{A}:\Sigma \to SL(2, \mathbb{R})$ be a one-step cocycle. Assume that $\mathcal{A}:\Sigma \to SL(2,\mathbb{R})$ is a typical cocycle. Then, the cocycle $\mathcal{A}$ is dominated with respect to the Mather set $\mathcal{K}$.

Figures (1)

  • Figure 1.1: This is an example of a uniformly hyperbolic $A=(A_{1},A_{2})$ and a multicone with with 5 components that satisfies the $\mathrm{NOC}$ condition. Inner arrows indicate stable and unstable directions of $A_{1}$ and $A_{2}$. Blue and red outer arrows indicate the action of $A_{1}$ and $A_{2}$ in the components of the multicone, respectively.

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Theorem 1.4
  • Example 1.5
  • Example 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • ...and 14 more