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Finitely Supported Measures on \(S{L}_{2}(\mathbb{R})\) Which are Absolutely Continuous at Infinity

Part of the Lecture Notes in Mathematics book series (LNM,volume 2050)

Abstract

We construct finitely supported symmetric probability measures on \(S{L}_{2}(\mathbb{R})\) for which the Furstenberg measure on \({\mathbb{P}}_{1}(\mathbb{R})\) has a smooth density.

Keywords

  • Probability Measure
  • Haar Measure
  • Projective Action
  • Absolute Continuity
  • Similar Construction

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. B. Barany, M. Pollicott, K. Simon, Stationary measures for projective transformations: The Blackwell and Furstenberg measures. Preprint (2010)

    Google Scholar 

  2. J. Bourgain, Expanders and dimensional expansion. C.R. Math. Acad. Sci. Paris 347(7–8), 356–362 (2000)

    Google Scholar 

  3. J. Bourgain, A. Gamburd, On the spectral gap for finitely generated subgroups of SU(2). Invent. Math. 171(1), 83–121 (2008)

    Google Scholar 

  4. V. Kaimanovich, V. Le Prince, Matrix random products with singular harmonic measure. Geom. Ded. 150, 257–279 (2011)

    Google Scholar 

  5. K. Simon, B. Solomyak, M. Urbanski, Hausdorff dimension of limit sets for parabolic IFS with overlap. Pac. J. Math. 201(2), 441–478 (2001)

    Google Scholar 

  6. K. Simon, B. Solomyak, M. Urbanski, Invariant measures for parabolic IFS with overlaps and random continued fractions. Trans. Am. Math. Soc. 353(12), 5145–5164 (2001)

    Google Scholar 

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Acknowledgements

The author is grateful to C. McMullen and P. Varju for several related discussions. Research was partially supported by NSF grants DMS-0808042 and DMS-0835373

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Correspondence to Jean Bourgain .

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© 2012 Springer-Verlag Berlin Heidelberg

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Bourgain, J. (2012). Finitely Supported Measures on \(S{L}_{2}(\mathbb{R})\) Which are Absolutely Continuous at Infinity. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_7

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