Exact Lyapunov exponents of the generalized Boole transformations

Abstract

The generalized Boole transformations have rich behavior ranging from the \textit{mixing} phase with the Cauchy invariant measure to the \textit{dissipative} phase through the \textit{infinite ergodic} phase with the Lebesgue measure. In this Letter, by giving the proof of mixing property for $0<α<1$ we show an \textit{analytic} formula of the Lyapunov exponents $λ$ which are explicitly parameterized in terms of the parameter $α$ of the generalized Boole transformations for the whole region $α>0$ and bridge those three phase \textit{continuously}. We found the different scale behavior of the Lyapunov exponent near $α=1$ using analytic formula with the parameter $α$. In particular, for $0<α<1$, we then prove an existence of extremely sensitive dependency of Lyapunov exponents, where the absolute values of the derivative of Lyapunov exponents with respect to the parameter $α$ diverge to infinity in the limit of $α\to 0$, and $α\to 1$. This result shows the computational complexity on the numerical simulations of the Lyapunov exponents near $α\simeq$ 0, 1.