Abstract.
We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversally conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts a ball exponentially. We deduce some results about the unique ergodicity.
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The authors acknowledge support from the Swiss National Science Foundation. The second author’s work was partially supported by grants RBFR 02-01-00482, RFBR 02-01-22002, CRDF RM1-2358-MO-02.
Received: June 2005, Revision: January 2006, Accepted: March 2006
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Deroin, B., Kleptsyn, V. Random Conformal Dynamical Systems. GAFA Geom. funct. anal. 17, 1043–1105 (2007). https://doi.org/10.1007/s00039-007-0606-y
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DOI: https://doi.org/10.1007/s00039-007-0606-y