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Expanding interval maps with intermittent behaviour, physical measures and time scales
We exhibit a new family of piecewise monotonic expanding interval maps
with interesting intermittent-like statistical behaviours. Among these maps, there
are uniformly expanding ones for which a Lebesgue-typical orbit spends most
of the time close to an "indifferent Cantor set" which plays the role of
the usual neutral fixed point. There are also examples with an indifferent
fixed point and an infinite absolutely continuous invariant measure.
Like in the classical case, the Dirac mass at $0$ describes the
statistical behaviour at usual time scale while the
infinite one tells about the statistical behaviour at larger scales.
But, here, there is another invariant measure describing
the statistical behaviour of the ergodic sums at a third natural (intermediate) time scale.
To try to understand this last phenomenon, we propose a more general
construction that yields an example for which we conjecture there is an
infinite number of natural time scales.