Families of piecewise linear maps with constant Lyapunov exponent

TL;DR

This paper investigates families of piecewise linear maps with varying slopes, demonstrating that in certain parameter regions, the Lyapunov exponent and topological entropy remain constant, supported by numerical and analytical evidence.

Contribution

It introduces new families of maps with constant Lyapunov exponents across parameter regions and provides both numerical and analytical validation of this phenomenon.

Findings

Lyapunov exponent remains constant in certain parameter regions

Topological entropy is also constant in these regions

The mechanism differs from logistic map dynamics

Abstract

We consider families of piecewise linear maps in which the moduli of the two slopes take different values. In some parameter regions, despite the variations in the dynamics, the Lyapunov exponent and the topological entropy remain constant. We provide numerical evidence of this fact and we prove it analytically for some special cases. The mechanism is very different from that of the logistic map and we conjecture that the Lyapunov plateaus reflect arithmetic relations between the slopes.