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A subgroup of Homeo + (S 1) is Möbius-like if every element is conjugate to an element of PSL (2 , R) . In general, a Möbius-like subgroup of Homeo + (S 1) is not necessarily (semi-)conjugate to a subgroup of PSL (2 , R) , as discovered by Kovačević (Trans Am Math Soc 351:4823–4835, 1999). Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of Homeo + (S 1) which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of PSL (2 , R) . On the other hand, we provide an example of elementary subgroup of Diff + ∞ (S 1) satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of PSL (2 , R) . Finally, we show that non-elementary, non-locally discrete subgroups acting with at most N fixed points are conjugate to a dense subgroup of some finite central extension of PSL (2 , R) . [ABSTRACT FROM AUTHOR]

This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite graphs: these are the so-called hierarchical graphs. They possess a well-defined level structure and any level is built using a simple recursion. Stopping the construction at any finite level, we have a discrete random metric space when we set the edges to have random length (using a multiplicative cascade with fixed law $${m}$$ ). We introduce a tool, the cut-off process, by means of which one finds that renormalizing the sequence of metrics by an exponential factor, they converge in law to a non-trivial metric on the limit space. Such limit law is stationary, in the sense that glueing together a certain number of copies of the random limit space, according to the combinatorics of the brick graph, the obtained random metric has the same law when rescaled by a random factor of law $${m}$$ . In other words, the stationary random metric is the solution of a distributional equation. When the measure m has continuous positive density on $${\mathbf{R}_{+}}$$ , the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation. We also investigate topological and geometric properties of the random space when m is log-normal, detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade. [ABSTRACT FROM AUTHOR]

The article focuses on the automorphic measures for circle diffeomorphisms based on theorems proving the mathematical feasibility of irrational angle, rotation number and invariant probability measure of ƒ. The authors provide proofs for difeomorphisms with a minimal invariant Cantor set by recalling diffeomorphism of irrational rotation number with invariant 1-distribution dissimilar from the invariant measure.