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
μ
σ
scaling at the fixed point at x=0 as
x
σ
, but smooth elsewhere. Here
σ
should be smaller than a critical value
σ
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.