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Critical states of transient chaos

Z. Kaufmann, A. Németh, P. Szépfalusy

chao-dynarXiv:chao-dyn/9907003

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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