2025/10/13 by Bhattacharyya, Anubrato, Banerjee, Kuntal
#37B55 #37D05 #37E10 #37G35 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.11441
Inspired by the 2007 work by M.~Misiurewicz and A.~Rodrigues [Double Standard Maps, M. Misiurewicz, A. Rodrigues, Communications in Mathematical Physics], we consider a family of circle maps that are perturbations of the doubling map on the circle by a piecewise linear map. We call this the piecewise linear perturbation of the doubling map (PLPDM) and it is given by the formula, fa,b(x)= (2x+a+\dfracb2 S(x) ) // 1 \text for x, a, b ∈ [0,1] , where y // 1 means y \mod 1 (or simply, the fractional part of y) and S(x) is the piecewise linear approximation of sin 2π(x-1/4). The map S(x) is called the straight sine map. Define the hyperbolic set, H= \ (a,b) ∈ ℝ/ℤ × [0,1] : fa,b has an attracting cycle \. Tongues are defined as the components of H that touch the ceiling \b=1\ in a non degenerate interval. Any other component is referred to as an Eye. We show the uniqueness of the attracting cycle of fa,b for (a,b) ∈ H. We then define type and prove the existence of the tongues of all types. We also show how combinatorics of the attracting orbit determines if the component is a tongue or an eye. We show that fa,b is conjugate to the doubling map if (a,b) ∉ H. Some experimental proof of the existence of eyes in the parameter space corresponding to different combinatorics will be shown.