Critical states of transient chaos

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.