Research — Dan Thompson

About

I am a Professor in the Mathematics Department of the Ohio State University, where I am a member of the Ergodic theory group. I joined Ohio State in 2012 (Assistant Professor 2012–2017, Associate Professor 2017–2020, Professor 2020–). From 2009–2012 I was Chowla Research Assistant Professor at Penn State, where I was a member of the Center for Dynamics and Geometry. I completed my PhD in Mathematics at the University of Warwick in 2009, supervised by Professors Mark Pollicott and Peter Walters. My research area is ergodic theory and dynamical systems. I am an organiser of the Ohio State ergodic theory seminar. My research is currently supported by the National Science Foundation.

I am an organizer of the conference New Directions in Thermodynamic Formalism at CIRM, Marseille, June 2026.

I currently advise two PhD students — Austin Allen and Tony Passero. Previous advisees who currently hold academic positions include Tianyu Wang (Assistant Professor, SJTU Shanghai), Ben Call (Assistant Professor, University of Wisconsin-Whitewater), and Caleb Dilsavor (postdoc, Northwestern). A list of my former students and their theses is here.

I am a Senator of the Ohio State University Senate (2021–2024, re-elected 2024–2026, re-elected 2026–2029) and currently chair of the Ohio State University Council for Enrollment and Student Progress.

I am an Editor of the journal Ergodic Theory and Dynamical Systems (Executive Editor 2023–2026, Managing Editor 2027–).

I'm a bassist. Some videos are in the music section below.

Last updated: October 2026.

Research interests

My main research area is ergodic theory and dynamical systems, particularly the "entropy theory" of this subject. I am particularly interested in connections between dynamical systems and other areas of mathematics. Recently my work has focused especially on connections between dynamical systems and geometry. I have also worked on the implementation of ideas from my pure research in the applied setting of bioinformatics.

Current grants

PI, NSF Standard Grant, DMS-2349915, Advances in Ergodic Geometry, 2024–2027.

Preprints

Available at my arXiv page.

Hölder regularity for metric Anosov flows

Joint work with Caleb Dilsavor.
34 pages.
arXiv
Thermodynamic formalism for non-compact systems with expansivity and specification

Joint work with Vaughn Climenhaga and Tianyu Wang.
96 pages.
arXiv
The specification approach to equilibrium states for parabolic rational maps

Joint work with Katelynn Huneycutt.
18 pages.
arXiv

A paper with Paschal and Somers coming soon. Some new work has been announced in talk abstracts — see the talks page.

Published papers and papers accepted for publication

Gibbs measures for geodesic flow on CAT(−1) spaces

Joint work with Caleb Dilsavor.
Advances in Mathematics, 2025, Vol. 469, 63 pages.
arXiv
Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows

Joint work with Benjamin Call.
Journal of the London Mathematical Society, 2022, Vol. 105(2), p.797–824. 26 pages.
arXiv
Fluctuations of time averages around closed geodesics in non-positive curvature

Joint work with Tianyu Wang.
Communications in Mathematical Physics, 2021, Vol. 385(2), p.1212–1243. 29 pages.
arXiv
Measures of maximal entropy on subsystems of topological suspension semi-flows

Joint work with Tamara Kucherenko.
Studia Mathematica, 2021, Vol. 260, p.229–240. 10 pages.
arXiv
Beyond Bowen's Specification Property

Survey paper joint with Vaughn Climenhaga.
72 pages. To appear in Springer Lecture Notes in Mathematics Series, CIRM Jean-Morlet Chair Subseries (Fall 2019 Semester).
arXiv
Strong symbolic dynamics for geodesic flow on CAT(−1) spaces and other metric Anosov flows

Joint work with Dave Constantine and Jean-François Lafont.
Journal de l'École polytechnique — Mathématiques (JEP), 2020, Vol. 7, p.201–231.
arXiv
Measures of maximal entropy for suspension flows over the full shift

Joint work with Tamara Kucherenko.
Mathematische Zeitschrift, 2020, Vol. 294, p.769–781.
arXiv
Unique equilibrium states for geodesic flows in nonpositive curvature

Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher.
GAFA, 2018, Vol. 28 issue 5, p.1209–1259. Correction to section 8 (PDF).
arXiv
The weak specification property for geodesic flows on CAT(−1) spaces

Joint work with Dave Constantine and Jean-François Lafont.
Groups, Geometry and Dynamics, 2020, Vol. 14 issue 1, p.297–336.
arXiv
Fat flats in rank one manifolds

Joint work with Dave Constantine, Jean-François Lafont and Ben McReynolds.
Michigan Mathematical Journal, 2019, Vol. 68 issue 2, p.251–275.
arXiv
Unique equilibrium states for Bonatti–Viana diffeomorphisms

Joint work with Vaughn Climenhaga and Todd Fisher.
Nonlinearity, 2018, Vol. 31 issue 6, p.2532–2577.
arXiv
Generalized beta-transformations and the entropy of unimodal maps

Commentarii Mathematici Helvetici, 2017, Vol. 92 issue 4, p.777–800.
arXiv
A suspension flow over the full shift with two distinct measures of maximal entropy

Joint work with Tamara Kucherenko.
Topology Proceedings, 2018, Vol. 52, p.321–328.
Unique equilibrium states for Mañé diffeomorphisms

Joint work with Vaughn Climenhaga and Todd Fisher.
Ergodic Theory and Dynamical Systems, 2019, Vol. 39 issue 9, p.2433–2455.
arXiv
Unique equilibrium states for flows and homeomorphisms with non-uniform structure

Joint work with Vaughn Climenhaga.
Advances in Mathematics, 2016, Vol. 303, p.745–799.
arXiv
Large deviations for systems with non-uniform structure

Joint work with Vaughn Climenhaga and Kenichiro Yamamoto.
Transactions of the American Mathematical Society, 2017, Vol. 369, p.4167–4192. Small correction needed.
arXiv
Coding sequence density estimation via topological pressure

Joint work with David Koslicki.
Journal of Mathematical Biology, 2015, Vol. 70 issue 1–2, p.45–69.
arXiv
Intrinsic ergodicity via obstruction entropies

Joint work with Vaughn Climenhaga.
Ergodic Theory and Dynamical Systems, 2014, Vol. 34 issue 6, p.1816–1831.
arXiv
Equilibrium states beyond specification and the Bowen property

Joint work with Vaughn Climenhaga.
Journal of the London Mathematical Society, 2013, Vol. 87 part 2, p.401–427.
arXiv
Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors

Joint work with Vaughn Climenhaga.
Israel Journal of Mathematics, 2012, Vol. 192 part 2, p.785–817. Small correction needed.
arXiv
Irregular sets, the beta-transformation and the almost specification property

Transactions of the American Mathematical Society, 2012, Vol. 364 no. 10, p.5395–5414.
arXiv
A criterion for topological entropy to decrease under normalised Ricci flow

Discrete and Continuous Dynamical Systems, Series A, 30(4), 2011.
arXiv
The irregular set for maps with the specification property has full topological pressure

Dynamical Systems: An International Journal, 25(1), 2010, p.25–51.
arXiv
A variational principle for topological pressure for certain non-compact sets

Journal of the London Mathematical Society, December 2009, Vol. 80 part 3, p.585–602.
arXiv
A thermodynamic definition of topological pressure for non-compact sets

Ergodic Theory and Dynamical Systems, 31(2), 2011, p.527–547.
arXiv
PhD Thesis — Warwick University, 2009

PDF

Exposition

Three short articles written for the Early Career section of the Notices of the American Mathematical Society can be found here. My most recent one is on a productivity theme: Do Mathematics Every Day.

I gave a series of four lectures for graduate students entering the field in Trieste, Italy, July 2024. The first video is here. The rest of the videos, notes, and problems can be found here.

A series of three lectures on the Katok Entropy Rigidity Theorem for negative curvature surfaces (from Math 7222 Ergodic Theory II, OSU, Spring 2021): Lecture 1, Lecture 2, Lecture 3, Notes (PDF).

Notes from mini-course at Beyond Uniform Hyperbolicity 2019: Beyond Bowen's Specification Property (II). Videos of the lectures are published by CIRM here.

Upcoming talks and travel

Workshop on Rigidity Theory of Diffeomorphisms and Flows, honoring Keith Burns's 70th birthday, Brin Mathematics Research Center, University of Maryland, 19–23 October 2026.

Recent talks

Talks from the last several years are listed below. For the full list and selected abstracts, see the talks page.

Workshop on Smooth Dynamical Systems, Wake Forest University, January 2026. Video
Dynamics seminar, Northwestern University, Chicago, December 2025
Dynamics seminar, University of Illinois Chicago, December 2025
Dynamics seminar, Graduate Center, City University of New York, October 2025
Conference on Dynamical Systems: Geometric and Statistical Properties, University of Warwick, UK, July 2024
Mini-course, Summer school on dynamics, ICTP, Trieste (Italy), July 2024
Penn State Workshop on Dynamical Systems, Session in honor of Todd Fisher, November 2023
Dynamics seminar, University of Houston, March 2023
Plenary lecture, Spring Topology Conference, Rhodes College (Virtual), March 2023
Dynamics seminar, Northwestern University, November 2022
Special session on dynamical systems of geometric origin, AMS Sectional Meeting, Tufts (Virtual), March 2022
Math Dept Colloquium (online), George Washington University, October 2021
Dynamics seminar (online), University of Toronto, November 2021
Dynamics seminar (online), Penn State, February 2021
Expanding Dynamics Meeting (online), Santiago, Chile, January 2021. Conference website
One-Day Ergodic Theory Meeting (online), United Kingdom, December 2020
Dynamics seminar (online), Resistencia Dinamica, Rio de Janeiro, Brazil, November 2020
Dynamics Seminar (online), University of Wisconsin–Madison, October 2020
Dynamics seminar, IUPUI, February 2020
Workshop on Statistical aspects of geodesic flows in nonpositive curvature, University of Warwick, January 2020
Mini-course (6 part) 'Beyond Bowen's specification property', joint with Vaughn Climenhaga, Beyond Uniform Hyperbolicity Conference, Luminy (France), May 2019
Speaker, Maryland Conference on Dynamical Systems, University of Maryland, April 2019
Participant/Organizer/Speaker, Workshop on Thermodynamic Formalism, AIM, San Jose CA, July 2019
Participant, Conference on Thermodynamic Formalism: Modern Techniques in Smooth Ergodic Theory, Luminy (France), July 2019
Participant/Organizer, Workshop on Thermodynamic Formalism: Ergodic theory and geometry, University of Warwick, July 2019
Speaker, Groups Geometry and Dynamics 1-day conference, Indianapolis, February 2019
Dynamics seminar, University of Houston, February 2019

Conferences and workshops organized

Conference on New Directions in Thermodynamic Formalism, CIRM, Luminy, France, 22–26 June 2026 — with Boris Hasselblatt, Jérôme Buzzi and Tamara Kucherenko — conference website
Conference in honor of Todd Fisher, Brigham Young University, June 11–13 2024 — with Keith Burns, Amie Wilkinson, Davi Obata, Ben Webb and Lennard Bakker
Workshop at BIRS-Okanagan (with Keith Burns, Tamara Kucherenko and Agnieszka Zelerowicz) on Thermodynamic Formalism and Geometry, Summer 2023
Organizer, Ohio State Ergodic Theory and Probability Seminar, 2012–2022
Workshop on best practices in Graduate Advising at Ohio State, June 2019 — with Angelica Cueto, David Penneys and Krystal Taylor
AIM workshop on 'Equilibrium states for dynamical systems arising from geometry', July 2019 — with Keith Burns, Vaughn Climenhaga and Todd Fisher
Workshop on dynamical systems Thermodynamic Formalism: Ergodic Theory and Geometry at Warwick, celebrating the 60th birthday of Mark Pollicott, July 2019 — with Oliver Jenkinson, Thomas Jordan and Richard Sharp
Organiser/mentor (with Jon Chaika, Vaughn Climenhaga, Boris Hasselblatt and Bryna Kra) for an AMS Mathematics Research Community
Workshop on Thermodynamic Formalism, ICMS, Edinburgh, June 2018 — conference website — with Thomas Jordan, Omri Sarig and Mike Todd
Midwest Dynamical Systems Meeting at Ohio State, October 2015 — conference website
Ohio State Young Scholars Program 2013, Mathematics director (with John Johnson and Adriana Dawes)
Organizer, Ohio State Math Department Colloquium, 2013–2014; Co-organiser 2016–2017
AMS special session on Dynamical Systems: Thermodynamic formalism and connections with geometry, Boulder CO, April 2013 — with Keith Burns
Organizer, Penn State Center for Dynamics and Geometry seminar, 2009–2011
Co-organiser (with Mark Pollicott) of Dynamical Systems seminar, University of Warwick, 2008–2009
Organizer, workshop Chaotic Properties of Dynamical Systems, University of Warwick, August 2007 — with Thomas Jordan

Former students

Equilibrium States for Parabolic Rational Maps

Katelynn Huneycutt · The Ohio State University, 2026
Thermodynamic formalism in coarse hyperbolicity

Caleb Dilsavor · The Ohio State University, 2024
Rosettes and Beta Numbers

Michael Lane · The Ohio State University, 2024
Uniqueness and Mixing Properties of Equilibrium States

Benjamin Call · The Ohio State University, 2022
Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems

Tianyu Wang · The Ohio State University, 2021

Past awards and grants

co-PI, NSF conference grant DMS-2230827 'Midwest Dynamical Systems Conferences 2022/2023' with Roland Roeder (PI) and Osama Khalil
PI, NSF grant DMS-1954463, 'New Directions in Thermodynamic Formalism for Geodesic Flows Beyond the Closed Riemannian Case', 2020–2024
co-PI, NSF conference grant DMS-1856176 'Midwest Dynamical Systems Conferences 2019/2020' with Laura DeMarco (PI) and Roland Roeder
PI, NSF CAREER award DMS-1461163, 'Entropy in dynamics: connections with geometry, algebraic numbers, and bioscience', 2015–2021
Participant, AIM SQuaRE with Keith Burns, Vaughn Climenhaga and Todd Fisher, 2013–2016 — "Equilibrium states for geodesic flows in nonpositive curvature"
NSF "standard" grant, 2011–2015
August 2007 — Winner (with Thomas Jordan), University of Warwick ASSEC Conference Competition to organise 'Chaotic Properties of Dynamical Systems'
August 2007 — London Mathematical Society Scheme 1 Conference Grant, 'Chaotic Properties of Dynamical Systems'