  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.
