Critical states of transient chaos

One-dimensional maps exhibiting transient chaos and defined on two preimage s of the unit interval [0,1] are investigated. It is shown that such maps h ave continuously many conditionally invariant measures mu(sigma) scaling at the fixed point at x=0 as x(sigma), but smooth elsewhere. Here, a should b e smaller than a critical value sigma(c) that is related to the spectral pr operties of the Frobenius-Perron operator. The corresponding natural measur es are proven to be entirely concentrated in the fixed point.