Critical states of transient chaos
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.
chao-dyn↗