On the pointwise Lyapunov exponent of holomorphic maps

TL;DR

This paper establishes conditions under which the lower Lyapunov exponent of holomorphic maps is non-negative for various types of orbits, extending understanding of stability and chaos in complex dynamics.

Contribution

It proves non-negativity of the lower Lyapunov exponent for bounded and unbounded orbits in holomorphic maps under broad conditions, including slow accumulation to infinity or singular sets.

Findings

Lower Lyapunov exponent is non-negative for bounded orbits avoiding singular sets or attracting cycles.

The result extends to unbounded orbits with bounded singular sets.

Orbits accumulating slowly to infinity or singular sets also have non-negative Lyapunov exponents.

Abstract

We prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.