Vaughn Climenhaga

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 information on my:

papers and preprints

books I've written

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:

• 2022-2025: NSF DMS-2154378, "Hyperbolicity with singularities and coexistence via smoothing"
• 2022-2023: A Simons Foundation Fellowship
  
• 2016-2023: 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.

1. Gibbs measures have local product structure
   Preprint, submitted.
   
   
2. Equilibrium measures for two-sided shift spaces via dimension theory
    - with Jason Day.
   Preprint, submitted.
   
   
3. Lyapunov exponents and nonadapted measures for dispersing billiards
    - with Mark Demers, Yuri Lima, and Hongkun Zhang.
   Communications in Mathematical Physics, 405, 24 (2024)
   
   
4. SRB and equilibrium measures via dimension theory
   in A Vision for Dynamics in the 21st Century: The Legacy of Anatole Katok (2024), pp. 94-138, Cambridge.
   
   
5. Closed geodesics on surfaces without conjugate points
    - with Gerhard Knieper and Khadim War.
   Communications in Contemporary Mathematics, 24 (2022) No. 06, 2150067.
   
   
6. SRB measures and Young towers for surface diffeomorphisms
    - with Stefano Luzzatto and Yakov Pesin.
   Annales Henri Poincaré, 23 (2022), 973-1059
   


7. Beyond Bowen's specification property
   (Survey) - with Daniel J. Thompson.
   in Thermodynamic Formalism (2021), pp. 3-82, Springer Lecture Notes in Mathematics Vol. 2290, CIRM Jean-Morlet Chair subseries
   
   • A small correction is needed in the proofs of two propositions.
     
   
   
8. 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, 376 (2021), 107452.
   
   
9. Equilibrium measures for some partially hyperbolic systems
    - with Yakov Pesin and Agnieszka Zelerowicz.
   Journal of Modern Dynamics, 16 (2020), 155-205.
   
   • The published version has a minor error in the proof of Lemma 6.6.
     
   
   
10. Positive entropy equilibrium states.
     - with Van Cyr.
    Israel Journal of Mathematics, 232 (2019), no. 2, 899-920.
    
    
11. One-sided almost specification and intrinsic ergodicity.
    - with Ronnie Pavlov.
    Ergodic Theory and Dynamical Systems, 39 (2019), 2456-2480.


12. Equilibrium states for Mañé diffeomorphisms.
     - with Todd Fisher and Daniel J. Thompson.
    Ergodic Theory and Dynamical Systems, 39 (2019), 2433-2455.
    
    
13. 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.
    
    
14. Specification and towers in shift spaces.
    Communications in Mathematical Physics, 364, issue 2 (2018), 441-504.
    
    
15. 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.


16. 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
      
      
17. 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.
    
    • Condition [A.1] in Theorems A and B needs to be strengthened slightly.
      
      
18. The geometric approach for constructing Sinai-Ruelle-Bowen measures
    (Survey) - with Stefano Luzzatto and Yakov Pesin. 
    Journal of Statistical Physics, 166 (2017), 467-493.
    
    
19. Building thermodynamics for non-uniformly hyperbolic maps
    (Survey) - with Yakov Pesin. 
    Arnold Mathematical Journal, 3 (2017), 37-82.
    
    
20. Unique equilibrium states for flows and homeomorphisms with non-uniform structure.
     - with Daniel J. Thompson.
    Advances in Mathematics, 303 (2016), 745-799.
    
    
21. 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.
    
    
22. Hadamard-Perron theorems and effective hyperbolicity.
     - with Yakov Pesin.
    Ergodic Theory and Dynamical Systems, 36 (2016), 23-63.
    
    
23. Intrinsic ergodicity via obstruction entropies.
     - with Daniel J. Thompson.
    Ergodic Theory and Dynamical Systems, 34 (2014), 1816-1831.
    
    
24. The thermodynamic approach to multifractal analysis.
    Ergodic Theory and Dynamical Systems, 34 (2014), 1409-1450.
    
    • Supersedes Multifractal formalism derived from thermodynamics, an earlier work with some of the same results that is now a permanent preprint
    
    
25. Topological pressure of simultaneous level sets.
    Nonlinearity 26 (2013), 241-268.
    
    
26. Equilibrium states beyond specification and the Bowen property.
     - with Daniel J. Thompson.
    Journal of the London Mathematical Society 87 (2013), 401-427.
    
    
27. Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors.
     - with Daniel J. Thompson. 
    Israel Journal of Mathematics, 192 (2012), 785-817.
    
    • Two corrections: one to Theorem B, one regarding S-gap shifts and almost specification


28. Bowen's equation in the non-uniform setting. 
    Ergodic Theory and Dynamical Systems 31 (2011), 1163-1182.
    
    
29. Multifractal formalism derived from thermodynamics for general dynamical systems.
    Electronic Research Announcements in Mathematical Sciences 17 (2010), 1-11.
    
    
30. A note on two approaches to the thermodynamic formalism.
    Discrete and Continuous Dynamical Systems 27 (2010), 995-1005.
    
    
31. Open problems in the theory of non-uniform hyperbolicity
    (Survey) - with Yakov Pesin.
    Discrete and Continuous Dynamical Systems, 27 (2010), 589-607.
    
    
32. Measure theory through dynamical eyes
    (Expository) - with Anatole Katok.
    Permanent preprint.
    
    

Books.

• From Groups to Geometry and Back
  - with Anatole Katok, 2017.  (List of corrections)
  
  
• Lectures on Fractal Geometry and Dynamical Systems
  - with Yakov Pesin, 2009.
  
  
• Lectures on Surfaces
  - with Anatole Katok, 2008.

• February 9, 2024, One World Dynamics Seminar (online)
  "Entropy as a dimension" (slides)
  
  
• January 18, 2024, Rice University (colloquium)
  "Growth in geometry and dynamics" (notes)
  
  
• May 14, 2023, Bedlewo, Poland (Thermodynamic Formalism: Non-additive Aspects and Related Topics)
  "Specification for non-compact systems"
  
  
• February 3, 2023, AIM (Big ideas in dynamics, online)
  "Specification and measures of maximal entropy" (Slides on Prezi) (PDF slides)
  
  
• November 4, 2022, Penn State (Workshop in dynamical systems and related topics)
  "Hausdorff measure and the MME for general shift spaces"
  
  
• September 19, 2022, University of Houston (dynamics seminar)
  "Counting closed geodesics"
  
  
• August 24, 2022, Rice University (seminar)
  "Counting closed geodesics"
  
  
• March 1, 2022, Pennsylvania State University (dynamics working seminar)
  "Local product structure, the Margulis argument, and large-scale geometry"
  
  
• February 28, 2022, Pennsylvania State University (dynamics seminar)
  "Caratheodory dimension constructions and the product structure of equilibrium measures"
  
  
• February 14, 2022, University of Victoria (online seminar)
  "Counting closed geodesics" (Slides)
  
  
• November 5, 2021, George Washington University (departmental colloquium, online)
  "Counting geodesics" (Slides)
  
  
• September 20, 2021, Ruhr-Universität Bochum (Hyperbolic and Symplectic Dynamics, remote talk)
  "Counting closed geodesics on surfaces without conjugate points" (Slides)
  
  
• April 30, 2021, University of Porto (online seminar)
  "Counting closed geodesics on surfaces without conjugate points" (Slides)
  
  
• June 23, 2020, University of Utah online working seminar in ergodic theory
  "Specification and the measure of maximal entropy"
  (Board notes)   (Notes on my blog, or in pdf)
  
  
• March 9, 2020, Ohio State
  "Non-adapted measures for billiards and other systems with singularities"
  
  
• December 10, 2019, CIRM, Marseille (Thermodynamic Formalism: Dynamical Systems, Statistical Properties and their Applications)
  "Closed geodesics and the measure of maximal entropy on surfaces without conjugate points"
  
  
• September 29, 2019, Penn State (Workshop in Dynamical Systems and Related Topics)
  "Counting closed geodesics on surfaces without conjugate points"
  
  
• July 26, 2019, University of Warwick (Thermodynamic Formalism: Ergodic Theory and Geometry)
  "Bowen-Margulis measure for geodesic flows" (Slides)
  
  
• July 15, 2019, AIM, San Jose (Equilibrium states for dynamical systems arising from geometry)
  "Equilibrium states and non-uniform specification" (Video)
  
  
• May 13-17, 2019, CIRM, Marseille (Beyond Uniform Hyperbolicity)
  "Beyond Bowen's specification property" (3 lectures).  (Videos)  (Notes)
  
  
• April 13, 2019, University of Maryland (Workshop on dynamical systems and related topics)
  "The measure of maximal entropy for geodesic flows without conjugate points".
  
  
• July 17, 2018, Ruhr-Universität Bochum.
  "Unique equilibrium states for geodesic flows in nonpositive curvature".
  
  
• June 22, 2018, ICMS, Edinburgh (Thermodynamic formalism in dynamical systems)
  "Leaf measures, dynamical dimensions, and equilibrium states".
  
  
• April 17, 2018, University of Massachussetts, Amherst.
  "Unique equilibrium states for geodesic flows in nonpositive curvature".
  
  
• October 6, 2017, Penn State (Workshop in dynamical systems and related topics)
  "Leaf measures via dynamical dimensions".
  
  
• August 4, 2017, University of British Columbia (Current trends in dynamical systems and the mathematical legacy of Rufus Bowen)
  "Unique equilibrium states for geodesic flows in nonpositive curvature". (Slides)
  
  
• March 28, 2017, Trinity University, San Antonio.
  "Stochastic behavior in deterministic dynamics". (Slides)
  
  
• March 20, 2017, Penn State.
  "Young towers for surface diffeomorphisms".
  
  
• February 20-24, 2017, CIRM, Marseille (Non-uniformly hyperbolic dynamical systems: coupling and renewal theory)
  "Young towers for surface diffeomorphisms".
  
  
• February 3, 2017, UH Grad Student Seminar.
  "Stochastic behavior in deterministic dynamics". (Slides)
  

• January 6, 2017, Joint Mathematics Meetings, Atlanta (AMS special session on Ergodic Theory and Dynamical Systems)
  "Specification and Markov properties in shift spaces".
  
  
• December 1, 2016, Australian National University (Advances in Ergodic Theory, Hyperbolic Dynamics & Statistical Laws)
  "Specification and Markov properties in shift spaces". (Slides)
  
  
• October 29, 2016, Penn State (Workshop in dynamical systems and related topics)
  "A direct proof of the entropy gap for rank 1 manifolds".
  
  
• October 9, 2016, University of Denver (AMS Western Sectional Meeting: Special session on zero dimensional dynamics)
  "Specification and Markov properties in shift spaces".  (Slides)
  
  
• September 7, 2016, Sam Houston State University colloquium.
  "Stochastic behavior in deterministic dynamics".  (Slides)
  
  
• July 8, 2016, ICTP, Trieste, Italy.
  "Equilibrium states for geodesic flow in non-positive curvature".
  (School on algebraic, geometric and probabilistic aspect of dynamical systems and control theory)
  
  
• April 25-29, 2016, ESI, Vienna.
  "Equilibrium states for geodesic flow in non-positive curvature".
  (Thermodynamic formalism and mixing)
  
  
• March 7-11, 2016, ICERM, Brown University.
  "Non-uniform specification properties, thermodynamic formalism, and towers".
  (Fractal geometry, hyperbolic dynamics, and thermodynamical formalism)
  
  
• January 12, 2016, Northwestern University.
  "SRB measures and Young towers for surface diffeomorphisms".
  
  
• January 11, 2016, University of Chicago.
  "SRB measures and Young towers for surface diffeomorphisms".
  
  
• October 23, 2015, Penn State. 
  (Workshop in Dynamical Systems and Related Topics)
  "Unique equilibrium states for geodesic flow in non-positive curvature".
  
  
• June 10-13, 2015, Porto, Portugal.  (AMS-EMS-SPM International Meeting: Special session on recurrence, mixing, and fluctuations: statistics of dynamical systems)
  "Specification, statistical properties, and towers".
  
  
• June 2, 2015, ICTP, Trieste, Italy.
  "Effective hyperbolicity and SRB measures".
  
  
• May 18, 2015, University of Alabama at Birmingham.  (Conference in memory of Nikolai Chernov)
  "Specification, statistical properties, and towers".
  
  
• May 7, 2015, University of Maryland.
  "Unique equilibrium states for geodesic flow in non-positive curvature".
  
  
• April 30, 2015.  University of Mississippi.
  "Thermodynamic formalism and uniform mixing properties".
  
  
• March 14, 2015, Michigan State University.  (AMS Central Sectional Meeting: Special session on smooth dynamical systems and ergodic theory)
  "Effective hyperbolicity and SRB measures". (Slides)
  
  
• January 11, 2015, Joint Mathematics Meetings, San Antonio.  (AMS special session on ergodic theory and dynamical systems)
  "Unique equilibrium states for some robustly transitive systems".  (Slides)
  
  
• September 15, 2014, University of Toronto.
  "Equilibrium states, non-uniform mixing, and towers"
  
  
• August 21, 2014, Ohio State University.
  "Tower constructions from specification properties"
  
  
• August 11, 2014, Banff, Alberta, Canada.
  (Recent Progress in Dynamical Systems and Related Topics)
  "Unique equilibrium states for some robustly transitive systems".  (Video)
  
  
• July 8, 2014, Madrid.
  (AIMS Conference on Dynamical Systems, Differential Equations and Applications: Special session on entropy and statistical properties for smooth dynamics)
  "Tower constructions and specification properties".  (Slides)
  
  
• June 2, 2014, University of North Texas.
  (Logic and Dynamics)
  "Specification, hyperbolicity, and towers".
  
  
• May 30, 2014, UFRJ, Rio de Janeiro.
  (Probability in Dynamics)
  "Specification, hyperbolicity, and towers".  (Notes)
  
  
• May 14-22, 2014, University of Houston.
  Multiple talks as part of UH Summer School in Dynamical Systems.
  
  
• April 12, 2014, University of Maryland.
  (Workshop on Dynamical Systems and Related Topics)
  "Tower constructions and specification properties".  (Notes)
  
  
• December 2, 2013, CIRM, Marseilles.
  (Hyperbolicity and dimension)
  "Non-uniform specification, thermodynamic formalism, and towers". (Slides)
  
  
• October 31, 2013, Indiana University, Bloomington.
  "SRB measures, unstable manifolds, and effective hyperbolicity".
  
  
• October 28, 2013, University of Houston (Dynamics seminar).
  "SRB measures, unstable manifolds, and effective hyperbolicity".
  
  
• October 8, 2013, Texas Christian University (Colloquium).
  "Thermodynamic formalism for dynamical systems".  (Slides)
  
  
• September 12, 2013, Ohio State University.
  "SRB measures, unstable manifolds, and effective hyperbolicity".  (Notes)
  
  
• August 8, 2013, Guanajuato, Mexico.
  (Mathematical Congress of the Americas: Special Session on Symbolic Dynamics).
  "Large deviations using non-uniform specification properties".  (Slides)
  
  
• July 11, 2013, Pontificia Universidad Católica de Chile, Santiago.
  (Thermodynamic formalism and applications).
  "Thermodynamics for discontinuous maps and potentials".  (Slides)
  
  
• May 13-21, 2013, University of Houston.
  Multiple talks as part of UH Summer School in Dynamical Systems.
  
  
• April 14, 2013, University of Colorado, Boulder.
  (AMS Western Sectional Meeting: Special Session on Dynamical Systems: Thermodynamic formalism and connections with geometry).
  "Large deviations and non-uniform specification properties".  (Slides)
  
  
• April 10, 2013, University of Houston (Undergraduate Colloquium).
  "The bigness of things". (Slides)
  
  
• April 8, 2013, University of Houston (Dynamics seminar).
  "Large deviations in dynamical systems".
  
  
• March 20, 2013, EPFL-CIB, Lausanne.
  (Workshop on Large deviations and thermodynamical formalism).
  "Non-uniform specification properties and large deviations". (Slides)
  
  
• March 3, 2013, University of Mississippi.
  (AMS Southeastern Sectional Meeting: Special Session on Dynamical Systems).
  "Effective hyperbolicity and applications of new Hadamard-Perron theorems".  (Slides)
  
  
• February 25, 2013, University of Houston.
  "The structure of the space of invariant measures".  (Notes)
  
  
• November 2, 2012, Rice University.
  "Sarig's construction of countable Markov partitions using Pesin theory".  (Notes)
  
  
• October 16, 2012, University of Houston (Pi Mu Epsilon).
  "Motivating examples in dynamical systems".  (Slides)
  
  
• October 5, 2012, University of Houston (Graduate student seminar).
  "Randomness and determinism in dynamical systems".  (Slides)