| The x-coordinate is the energy x=E. The y coordinate fixes the parameter y which is related to the magnetic flux. For rational y=p/q we can form f(x,y) = log[ | det (L(y) -x) | ]/q, where L(y) is a Jacobi matrix with side diagonal entries 1 and diagonal entries 2 cos(2 pi y*k). The function f(x,y) extends from rational y=p/q to all y. The Hofstadter butterfly is the set, where f(x,y)=0. The picture to the right is colored according to the value of f(x,y). The function f(x,y) is a Lyapunov exponent and defined for all x,y. |  |