On the size of the resonant set for the products of 2 × 2 matrices

Allen, Jeffrey; Seeger, Benjamin; Unger, Deborah

doi:10.2140/involve.2011.4.157

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On the size of the resonant set for the products of $2\times 2$ matrices

Jeffrey Allen, Benjamin Seeger and Deborah Unger

Vol. 4 (2011), No. 2, 157–166

DOI: 10.2140/involve.2011.4.157

Abstract

For $𝜃 \in [0, 2 π)$ and $λ > 1$ , consider the matrix $h = (\binom{λ}{0} \binom{0}{0})$ and the rotation matrix $R_{𝜃}$ . Let $W_{n} (𝜃)$ denote some product of $m$ instances of $R_{𝜃}$ and $n$ of $h$ , with the condition $m \leq 𝜖 n$ ( $0 < 𝜖 < 1$ ). We analyze the measure of the set of $𝜃$ for which $∥ W_{n} (𝜃) ∥ \geq λ^{δ n}$ ( $0 < δ < 1$ ). This can be regarded as a model problem for the Bochi–Fayad conjecture.

Keywords

Bochi–Fayad conjecture, resonant set, measure, rotation matrix, Fayad, Krikorian, exponential growth

Mathematical Subject Classification 2010

Primary: 37H15

Secondary: 37H05, 37C85

Milestones

Received: 10 December 2010

Revised: 17 February 2011

Accepted: 3 April 2011

Published: 17 January 2012

Communicated by Chi-Kwong Li

Authors

Jeffrey Allen
University Of Wisconsin – Madison Madison, WI 53706 United States

Benjamin Seeger
University Of Wisconsin – Madison Madison, WI 53706 United States

Deborah Unger
University Of Wisconsin – Madison Madison, WI 53715 United States

Vol. 4, No. 2, 2011