Dynamical systems: Hyperbolicity, invariant foliations, geodesic flows,
contact flows, and related topics
research, undertaken with colleagues from
several continents, is in the modern theory of
dynamical systems, with an emphasis on hyperbolic phenomena and on
geometrically motivated systems. I also write expository articles, write and
organize conferences and schools. Information about my publications can be
MathSciNet by those at an institution with a subscription. Among
my 5000 published pages is Introduction to the Modern Theory of Dynamical
Systems, one of the most cited mathematics books.
Former doctoral students of mine can be found in academic
positions at the Northwestern University, the University of Michigan, the
University of New Hampshire, and Queen's University.
While studying physics at the Technische Universität Berlin, I was awarded a Fulbright
Scholarship, which led to me obtaining an M.A. in mathematics from the
University of Maryland in College Park. I have been at Tufts University since
completing my doctorate at the California Institute of Technology in 1989.
During research leaves I have multiple times held visiting appointments at the Institut des
Hautes Etudes Scientifiques near Paris, the Institut de Recherche Mathématique
Avancée and the Collège Doctoral Européen in Strasbourg, the Eidgenössische
Technische Hochschule in Zürich, and the Graduate School of Mathematical
Sciences of the University of Tokyo. I have also held the Jean Morlet Chair at
the Centre International de Rencontres Mathématiques in Marseille, and I have
served within the governance of the American Mathematical Society, the American
Statistical Association and the Mathematical Association of America,
and on the Steering Committee of the Boston Higher Education
I serve as founding editor of the Journal of Modern Dynamics (from
2007), Founding Editor
in Chief of Mathematics Research Reports (from 2020), and as
Secretary of the American Mathematical Society (from 2021). In the Faculty of Arts,
Sciences and Engineering at Tufts I have chaired numerous standing faculty
committees as well as committees created for accreditation and for strategic
planning. My administrative roles at Tufts University have been chair of the
Department of Mathematics and Associate Provost.
Dynamical systems is the mathematical theory of
systems, such as in classical dynamics, that evolve in time. The motivations for
this field come from classical and celestial mechanics as well as, more
recently, from population dynamics, meteorology, economics, physiology,
neuroscience, medicine, genomics, i.e. all across the natural and social
sciences. In the modern theory of these systems the central aim is often to
answer qualitative questions about long-term behavior directly from a study of
the governing laws rather than through explicit expressions of the evolution
itself. The grandest of these questions is whether the solar system is stable
(or whether instead, the earth might, without external influences, leave the
solar system for good at some point). Just over 100 years ago this question
started the modern theory of dynamical systems and led to the first glimpse of
"chaos", and it led to another quantum leap, the KAM-theory of "order", in the
1950s. "Chaos theory" is a popular name for the study of dynamics in which the
cumulative effects of the tiniest discrepancies grow exponentially over time and
give rise to behavior that looks random and unpredictable. (If two people with
different calculators start with \(x=0.3\), compute
\(4 \cdot x \cdot (1-x)\), and repeatedly apply the same
formula to the output of the previous step, they will likely quite soon find
wild mismatches between their results due to accumulated deviations from
differences in rounding.) Hyperbolic dynamics is the mathematical study of these
systems, and much of my work has been in uniformly hyperbolic dynamics, which
represents this sensitivity to initial conditions in the purest and strongest
way and has been called the crown jewel of dynamical systems.
For a pure mathematician this is a beautiful subject to study because in these systems the
truly messy long-term behavior of any particular time evolution coexists with
smooth and orderly global structures in the space of possible states whose study
on one hand provides a way of understanding the possible long-term behaviors and
on the other hand can provide insights into the origins of the system, such as,
whether it arises from an algebraic system. It is an exciting field because in addition to intrinsic beauty it provides interactions with other
fields of mathematics (mostly differential geometry, but also number theory and
coding theory, for example). Moreover, the study of hyperbolic dynamics blends
into the study of real-world systems with complex behavior, providing methods
and concepts to the sciences for describing, classifying, studying and
understanding chaotic behavior, such as the intrinsic difficulty of weather
prediction or the diagnosis of an impending heart attack from the decreasing (!)
complexity of a patient's heart beat, but also the exquisitely subtle design of
chaotic trajectories for recent spacecraft missions.
Regularity of the Anosov splitting
years this project studied properties of the characteristic geometric structures
in hyperbolic dynamical systems notably in relation to geodesic flows, rigidity,
and fractal dimension. My coauthors are Gerber, Fang, Foulon, Keesing, Schmeling,
and Wilkinson. The resulting publications included:
- "A bootstrap for regularity of the Anosov splitting"
- "Anosov obstructions in higher dimension"
- "Horosperic foliations and relative pinching"
- "Periodic bunching and invariant foliations"
- "Regularity of the Anosov splitting and of Horospheric foliations"
- "Regularity of the Anosov splitting II"
- "Prevalence of non-Lipschitz Anosov foliations"
- "Critical regularity of invariant foliations"
- "The Riccati equation: Pinching of forcing and solutions"
- "Zygmund foliations"
- "Dimension product structure of hyperbolic sets"
- "Zygmund strong foliations in higher dimension"
- "Lipschitz-continuous invariant forms for algebraic Anosov systems"
- "Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows"
invariants for maximal isotropic C2 foliations" with Patrick Foulon
studies a remarkable geometric invariant that holds promise for new approaches
in smooth and geometric rigidity theory.
Ergodic theory of hyperbolic systems
In the 1980s
and again in 2013-14 I studied properties of invariant measures in hyperbolic
dynamics, first came "A Construction of the Margulis measure for Anosov flows,"
and recently, with Yves Coudène and Serge Troubetzkoy, "Multiple mixing from
hyperbolicity by the Hopf argument."
In the main
this involved a study of entropy theory with Zbigniew Nitecki and James Propp
that resulted in the publication "Topological entropy for nonuniformly
continuous maps," which shows how much the classically equivalent definitions
diverge when one goes beyond uniform continuity. With Keith Burns I revisited a
famous classical insight for one-dimensional maps in "The Sharkovsky Theorem: A
natural direct proof."
Exotic Anosov flows
Foulon I constructed exotic (i.e., nonalgebraic) Anosov flows with the contact
property ("Contact Anosov flows on hyperbolic 3-manifolds") and with Anne Vaugon
we are studying their periodic orbit growth more closely ("Periodic orbits of
contact Anosov flows on hyperbolic 3-manifolds"); Thomas Barthelmé and coauthors
have been studying the periodic orbit structure much more deeply yet.
Pointwise hyperbolicity implies uniform hyperbolicity
with Yakov Pesin and Jörg Schmeling is particularly interesting for the way in
which it connects descriptive set theory with dynamical systems.
Differentiability of the Hartman–Grobman Linearization
with Misha Guysinsky and Victoria Rayskin showed in that near a hyperbolic fixed
point the phase portrait looks geometrically just like the linearized one.
Degree-growth of monomial maps
with James Propp was my sole foray into algebraic dynamics.
Doctoral dissertations supervised:
John David Cowan "A billiard model for a gas of particles with rotation" (2004)
dissertation was the first work to study a gas of hard objects that are not
required to be spherical and resulted in the publications "A billiard model for
a gas of particles with rotation" and "Rigid particle systems and their billiard
models." The first publication shows that these systems can be modeled as a
point billiard, and the second one establishes that the boundary of that point
billiard is such as to make the study of the dynamics intractable.
Aaron Brown "Rigid properties of measures on the torus: smooth stabilizers and
This produced astonishing rigidity results for the structure of hyperbolic measures
as well as tools that have since led to significant progress in the
Zimmer program. The publications from this time
are "Constraints on dynamics preserving certain hyperbolic sets," "Nonexpanding
attractors: Conjugacy to algebraic models and classification in 3-manifolds,"
and "Smooth stabilizers for measures on the torus."
Thomas Barthelmé "A new Laplace operator in Finsler geometry and periodic orbits
of Anosov flows" (2012)
jointly supervised with Patrick Foulon, included two distinct remarkable
projects connected by Anosov flows. Among the resulting publications are "A
natural Finsler-Laplace operator" and "Knot theory of
Anosov flows: homotopy versus isotopy of closed orbits".
Donald Plante "On the interior of fat Sierpiński gaskets" (2012)
This led to
the eponymous publication which provided new ranges of contraction parameters
for which these gaskets have nonempty interior.
Sarah Bray "Dynamics of low-regularity Hilbert geometries" (2016)
This is a
remarkable case study of manifolds that are not Riemannian (but Finsler),
nonpositively curved but not CAT(0), and whose geodesic flows are nonuniformly
hyperbolic but not C1 nor volume-preserving. The toolkit developed
for this purpose will support not only deeper study of this subject but
investigations well beyond it.
Erica Clay "Desingularizing singular-hyperbolic attractors" (2016)
This project, initiated by Enrique Pujals, provides proof of concept for a strategy
to desingularize singular-hyperbolic attractors by a local surgery.