Vol. 4, No. 2, 2011

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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
Abstract

For θ [0,2π) and λ > 1, consider the matrix h =(λ 0 0 0) and the rotation matrix Rθ. Let Wn(θ) denote some product of m instances of Rθ and n of h, with the condition m ϵn (0 < ϵ < 1). We analyze the measure of the set of θ for which Wn(θ) λδn (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