On the Size of the Resonant Set for the Products of 2x2 Matrices

TL;DR

This paper investigates the measure of angles for which products of specific 2x2 matrices grow exponentially, providing insights into a model related to the Bochi-Fayad conjecture.

Contribution

It offers a measure estimate for the set of parameters where matrix products exhibit exponential growth, advancing understanding of the Bochi-Fayad conjecture.

Findings

Quantifies the measure of { heta} with exponential growth in matrix products.

Provides bounds related to the Bochi-Fayad conjecture.

Analyzes the size of the resonant set for matrix products.

Abstract

For {\theta} \in [0, 2{\pi}), consider the rotation matrix R? and h = ({\lambda}, 0; 0, 0), {\lambda} > 1. Let W_n({\theta}) denote the product of m R?'s and n h's with the condition m \leq [\epsilon\astn], (0 < \epsilon < 1). We analyze the measure of the set of {\theta} for which ||W_n({\theta})|| \geq {\lambda}?^(n*{\delta}), (0 < {\delta} < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.