  The crucial aspect of this demonstration is the discovery of renewal events,
hidden in the computed dynamics of a multifractal metronome, which enables the
replacement of the phenomenon of strong anticipation with a time delayed
cross-correlation between the driven and the driving metronome. We establish
that the phenomenon of complexity matching, which is the theme of an increasing
number of research groups, has two distinct measures. One measure is the
sensitivity of a complex system to environmental multi-fractality; another is
the level of information transfer, between two complex networks at criticality.
The cross-correlation function is evaluated in the ergodic long-time limit, but
its delayed maximal value is the signature of information transfer occurring in
the non ergodic short-time regime. It is shown that a more complex system
transfers its multifractality to a less complex system while the reverse case
is not possible.
