Flexibility of exponents for expanding maps on a circle

TL;DR

This paper investigates the range of Lyapunov exponents for smooth expanding maps on a circle of degree 2, demonstrating that the known bounds are the only restrictions on these exponents.

Contribution

It proves that the known bounds on Lyapunov exponents for degree 2 expanding maps are the only restrictions, clarifying the flexibility of these exponents.

Findings

Lyapunov exponents are bounded between 0 and log 2.

Equalities occur only when the exponents are at the bounds.

The bounds are the only restrictions on the exponents.

Abstract

We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to log 2 which, in addition, is less than or equal to the Lyapunov exponent of g with respect to the measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree 2.