Vaughn Climenhaga

Associate 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"

See here for a list of known corrections to the material below.


Papers.  Survey/expository papers are marked; all others are research papers.
  1. SRB and equilibrium measures via dimension theory
    Preprint, submitted.

  2. Beyond Bowen's specification property
    (Survey) - with Daniel J. Thompson.
    Preprint, submitted.

  3. Closed geodesics on surfaces without conjugate points
     - with Gerhard Knieper and Khadim War.
    Preprint, submitted.

  4. SRB measures and Young towers for surface diffeomorphisms
     - with Stefano Luzzatto and Yakov Pesin.
    Preprint, submitted.

  5. Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points
     - with Gerhard Knieper and Khadim War.
    Advances in Mathematics, to appear.

  6. Equilibrium measures for some partially hyperbolic systems
     - with Yakov Pesin and Agnieszka Zelerowicz.
    Journal of Modern Dynamics, 16 (2020), 155-205.

  7. Positive entropy equilibrium states.
     - with Van Cyr.
    Israel Journal of Mathematics, 232 (2019), no. 2, 899-920.

  8. One-sided almost specification and intrinsic ergodicity.
    - with Ronnie Pavlov.
    Ergodic Theory and Dynamical Systems
    , 39 (2019), 2456-2480.

  9. Equilibrium states for Mañé diffeomorphisms.
     - with Todd Fisher and Daniel J. Thompson.
    Ergodic Theory and Dynamical Systems, 39 (2019), 2433-2455.

  10. Equilibrium states in dynamical systems via geometric measure theory
    (Survey) - with Yakov Pesin and Agnieszka Zelerowicz.
    Bulletin of the American Mathematical Society, 56 (2019), no. 4, 569-610.

  11. Specification and towers in shift spaces.
    Communications in Mathematical Physics, 364, issue 2 (2018), 441-504.

  12. Unique equilibrium states for geodesic flows in nonpositive curvature.
     - with Keith Burns, Todd Fisher, and Daniel J. Thompson.
    Geometric and Functional Analysis, 28 (2018), no. 5, 1209-1259.

  13. Unique equilibrium states for Bonatti-Viana diffeomorphisms.
     - with Todd Fisher and Daniel J. Thompson.
    Nonlinearity, 31 (2018), no. 6, 2532-2570.
      • An earlier version also studied the Mañé examples; these were removed from the final version and now appear in their own paper

  14. 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.
  15. The geometric approach for constructing Sinai-Ruelle-Bowen measures
    (Survey) - with Stefano Luzzatto and Yakov Pesin. 
    Journal of Statistical Physics, 166 (2017), 467-493.

  16. Building thermodynamics for non-uniformly hyperbolic maps
    (Survey) - with Yakov Pesin. 
    Arnold Mathematical Journal, 3 (2017), 37-82.

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

  18. 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.

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

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

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

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

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

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

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

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

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

  28. Open problems in the theory of non-uniform hyperbolicity
    (Survey) - with Yakov Pesin.
    Discrete and Continuous Dynamical Systems, 27 (2010), 589-607.

  29. Measure theory through dynamical eyes
    (Expository) - with Anatole Katok.
    Permanent preprint.


Books.


Talks.  (Past and upcoming)


2020 Houston Workshop on Hyperbolic Dynamical Systems

2017 Mathematics Research Communities at Snowbird, Utah

2019 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