Vaughn Climenhaga

Assistant Professor
Department of Mathematics
University of Houston


My research interests are in dynamical systems and ergodic theory, particularly non-uniform hyperbolicity, thermodynamic formalism, dimension theory, and multifractal analysis.  Here you can find my papers and preprints, information on books I've written, and slides for talks I've given.

I also maintain a research blog, where I occasionally post about things that I find interesting but that for one reason or another don't fit (yet) into anything else I might write.  PDFs of some posts are linked to at right.

List of publications also available via MathSciNet or BibServer (not updated as frequently).

My research is supported by the following grants:

  • 2016-2021: NSF DMS-1554794, "CAREER: Unifying approaches to non-uniform hyperbolicity"
  • 2014-2017: NSF DMS-1362838, "Thermodynamics and statistics of non-uniformly hyperbolic dynamical systems"

Research papers.
  1. Positive entropy equilibrium states.
     - with Van Cyr.
    Preprint, submitted.

  2. Unique equilibrium states for geodesic flows in nonpositive curvature.
     - with Keith Burns, Todd Fisher, and Daniel J. Thompson.
    Preprint, submitted.

  3. One-sided almost specification and intrinsic ergodicity.
     - with Ronnie Pavlov.
    Preprint, submitted.

  4. Unique equilibrium states for Bonatti-Viana diffeomorphisms.
     - with Todd Fisher and Daniel J. Thompson.
    Preprint, submitted.
    • An earlier version also studied the Mañé examples; these have been removed from the current version and now appear in their own paper

  5. Equilibrium states for Mañé diffeomorphisms.
     - with Todd Fisher and Daniel J. Thompson.
    Preprint, submitted.

  6. Specification and towers in shift spaces.
    Preprint, submitted.

  7. Large deviations for systems with non-uniform structure.
    - with Daniel J. Thompson and Kenichiro Yamamoto.
    Transactions of the American Mathematical Society, 369 (2017), no. 6, 4167-4192.

  8. Unique equilibrium states for flows and homeomorphisms with non-uniform structure.
     - with Daniel J. Thompson.
    Advances in Mathematics, 303 (2016), 745-799.

  9. Non-stationary non-uniform hyperbolicity: SRB measures for non-uniformly hyperbolic attractors.
     - with Dmitry Dolgopyat and Yakov Pesin. 
    Communications in Mathematical Physics, 346, issue 2 (2016), 553-602.

  10. Hadamard-Perron theorems and effective hyperbolicity.
     - with Yakov Pesin.
    Ergodic Theory and Dynamical Systems, 36 (2016), 23-63.

  11. Intrinsic ergodicity via obstruction entropies.
     - with Daniel J. Thompson.
    Ergodic Theory and Dynamical Systems, 34 (2014), 1816-1831.

  12. The thermodynamic approach to multifractal analysis.
    Ergodic Theory and Dynamical Systems, 34 (2014), 1409-1450.

  13. Topological pressure of simultaneous level sets.
    Nonlinearity 26 (2013), 241-268.

  14. Equilibrium states beyond specification and the Bowen property.
     - with Daniel J. Thompson.
    Journal of the London Mathematical Society 87 (2013), 401-427.

  15. Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors.
     - with Daniel J. Thompson. 
    Israel Journal of Mathematics, 192 (2012), 785-817.

  16. Bowen's equation in the non-uniform setting
    Ergodic Theory and Dynamical Systems 31 (2011), 1163-1182.

  17. Multifractal formalism derived from thermodynamics for general dynamical systems.
    Electronic Research Announcements in Mathematical Sciences
    17 (2010), 1-11.

  18. A note on two approaches to the thermodynamic formalism.
    Discrete and Continuous Dynamical Systems 27 (2010), 995-1005.


Books.


Expository/survey papers.

Talks.  (Past and upcoming)


2017 Mathematics Research Communities at Snowbird, Utah

2017 Houston Summer School on Dynamical Systems



Selected expository blog posts
(from seminar lectures in preparation for our 2013 summer school)

Spectral methods in dynamics

Markov chains and mixing times

  • Part 1 (Feb. 11 lecture by Matt Nicol)
  • Part 2 (Feb. 18 lecture by Matt Nicol)
  • Part 3 (Feb. 25 lecture by Matt Nicol)

Convex cones and the Hilbert metric

Martingale methods