Critical states of transient chaos
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 μσ scaling at the fixed point at x=0 as xσ, but smooth elsewhere. Here σ should be smaller than a critical value σc 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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