Exact Lyapunov exponents of the generalized Boole transformations

The generalized Boole transformations have rich behavior ranging from the mixing phase with the Cauchy invariant measure to the dissipative phase through the infinite ergodic phase with the Lebesgue measure. In this letter, by giving the proof of mixing property for |0<α <1|⁠, we show an analytic formula of the Lyapunov exponents |λ| that are explicitly parameterized in terms of the parameter |α| of the generalized Boole transformations for the whole region |α >0| and bridge these three phases continuously. We find different scale behaviors of the Lyapunov exponent near |α =1| using an analytic formula with the parameter |α|⁠. In particular, for |0<α <1|⁠, we then prove the existence of the extremely sensitive dependence of Lyapunov exponents, where the absolute values of the derivative of the Lyapunov exponents with respect to the parameter |α| diverge to infinity in the limits of |α → 0| and |α → 1|⁠. This result shows the computational complexity of the numerical simulations of the Lyapunov exponents near |α 0, 1|⁠.