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 $x=0\mathrm{}$ 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 in the fixed point.

• Received 27 July 1999

DOI:https://doi.org/10.1103/PhysRevE.61.2543