Markov Extensions and Conditionally Invariant Measures for Certain Logistic Maps with Small Holes
Abstract
We study the family of quadratic maps f_a(x) = 1  ax^2 on the interval [1,1] with a between 0 and 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a density which is bounded away from zero and is analogous to the density for the corresponding closed system. These results establish the exponential escape rate of Lebesgue measure from the system, despite the contraction in a neighborhood of the critical point of the map. We also prove convergence of the conditionally invariant measure to the SRB measure for f_a as the size of the hole goes to zero.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404256
 Bibcode:
 2004math......4256D
 Keywords:

 Mathematics  Dynamical Systems;
 37E05 (primary) 37C40;
 37D25 (secondary)
 EPrint:
 35 pages. New version contains mainly minor revisions with some clarification of the arguments