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.
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