Temporal scaling at Feigenbaum points and non-extensive thermodynamics

TL;DR

This paper critically examines claims linking non-extensive thermodynamics to the Feigenbaum point in the logistic map, clarifying misconceptions and deriving new scaling laws based on attractor structure.

Contribution

It refutes recent claims connecting non-extensive thermodynamics to the Feigenbaum point and derives new scaling laws using attractor properties.

Findings

No generalized Pesin identity at the Feigenbaum point

Claims of non-stationary behavior are incorrect or misinterpreted

New scaling laws for the Feigenbaum attractor derived

Abstract

We show that recent claims for the non-stationary behaviour of the logistic map at the Feigenbaum point based on non-extensive thermodynamics are either wrong or can be easily deduced from well-known properties of the Feigenbaum attractor. In particular, there is no generalized Pesin identity for this system, the existing "proofs" being based on misconceptions about basic notions of ergodic theory. In deriving several new scaling laws of the Feigenbaum attractor, thorough use is made of its detailed structure, but there is no obvious connection to non-extensive thermodynamics.