Exact Lyapunov exponents of the generalized Boole transformations

TL;DR

This paper derives an explicit analytic formula for the Lyapunov exponents of generalized Boole transformations across different phases, revealing sensitive dependency on the parameter and complex behavior near critical points.

Contribution

It provides the first explicit analytic formula for Lyapunov exponents of generalized Boole transformations, covering all parameter regions and connecting different dynamical phases.

Findings

Explicit formula for Lyapunov exponents in terms of parameter .

Identification of scale behavior near  and sensitivity at  and .

Demonstration of computational complexity in numerical estimation near critical parameters.

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 $0<\alpha<1$ we show an \textit{analytic} formula of the Lyapunov exponents $\lambda$ which are explicitly parameterized in terms of the parameter $\alpha$ of the generalized Boole transformations for the whole region $\alpha>0$ and bridge those three phase \textit{continuously}. We found the different scale behavior of the Lyapunov exponent near $\alpha=1$ using analytic formula with the parameter $\alpha$. In particular, for $0<\alpha<1$, 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 $\alpha$ diverge to infinity in the limit of $\alpha\to 0$, and $\alpha \to 1$. This result shows the computational complexity on the numerical simulations of the Lyapunov exponents near $\alpha \simeq$ 0, 1.