Absolutely continuous Furstenberg measures for finitely-supported random walks
F\'elix Lequen
TL;DR
This paper extends Bourgain's construction of finitely-supported symmetric measures with smooth Furstenberg measures from SL(2,R) to general simple Lie groups, using harmonic analysis instead of Fourier series.
Contribution
It generalizes the construction of measures with smooth Furstenberg measures to all simple Lie groups, broadening the scope of Bourgain's original work.
Findings
Construction of finitely-supported measures with smooth Furstenberg measures for general simple Lie groups
Use of harmonic analysis on maximal compact subgroups instead of Fourier series
Extension of Bourgain's results beyond SL(2,R)
Abstract
In this note, we generalise a Bourgain's construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of to that of a general simple Lie group. The proof is the same as Bourgain's, except that the use of Fourier series is replaced by harmonic analysis on a maximal compact subgroup.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical Dynamics and Fractals · Geometric and Algebraic Topology · Markov Chains and Monte Carlo Methods