Multifractal invariant measures in expanding piecewise linear coupled maps

TL;DR

This paper investigates how coupling two expanding piecewise linear maps induces multifractal invariant measures, revealing complex behaviors that challenge existing multifractal theories.

Contribution

It demonstrates that coupling simple expanding maps results in multifractal measures, highlighting phenomena not explained by current theories.

Findings

Coupling induces multifractality in invariant measures.

Multifractal spectrum varies with coupling and map parameters.

Some spectra are robust, others vary significantly.

Abstract

We analyze invariant measures of two coupled piecewise linear and everywhere expanding maps on the synchronization manifold. We observe that though the individual maps have simple and smooth functions as their stationary densities, they become multifractal as soon as two of them are coupled nonlinearly even with a small coupling. For some maps, the multifractal spectrum seems to be robust with the coupling or map parameters and for some other maps, there is a substantial variation. The origin of the multifractal spectrum here is intriguing as it does not seem to conform to the existing theory of multifractal functions.