Critical states of transient chaos

Abstract

One-dimensional maps exhibiting transient chaos and defined on two preimages of the unit interval [0,1] are investigated. It is shown that such maps have continuously many conditionally invariant measures ${\mu }_{\sigma }$ scaling at the fixed point at x=0 as $x^{\sigma }$, but smooth elsewhere. Here $\sigma$ should be smaller than a critical value ${\sigma }_c$ that is related to the spectral properties of the Frobenius-Perron operator. The corresponding natural measures are proven to be entirely concentrated on the fixed point.

8 pages REVTeX file with 5 included figures; changes: added figures and remarks

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