My primary area of research is Dynamical Systems, a branch of modern mathematics concerned with time evolutions of natural and iterative processes. Earlier on in my career, my work revolved around the mathematical theory of chaos, focusing on topics such as Lyapunov exponents, entropy, fractal dimension, rates of mixing and strange attractors. A recurring theme was that deterministic, chaotic systems produce statistics much like those produced by random processes. I have also worked with concrete models, such as particle systems (billiards) and kicked oscillators.
In the last 10+ years, my interests shifted to dynamical systems that are large, random, and out of equilibrium. Two examples of large systems are networks of smaller systems and semiflows on function spaces defined by PDEs. I work with systems with both deterministic and stochastic components, as well as problems from Nonequilibrium Statistical Mechanics.
I have also developed a serious interest in a particular complex dynamical system, the brain. A significant part of my current research is in Theoretical Neuroscience.
Highlights of Research in Dynamical Systems
Entropy, Lyapunov exponents, and fractal dimension
Entropy and Lyapunov exponents are two different ways to capture dynamical complexity: entropy measures randomness in the sense of information theory, while positive Lyapunov exponents measure the rates at which nearby orbits diverge. It was known (shortly before I entered the field) that entropy is bounded above by the sum of positive Lyapunov exponents, and that these two quantities are equal for conservative systems. Ledrappier and I completed and clarified the picture: (A) We gave a necessary and sufficient condition for these two quantities to be equal, namely when the measure is SRB (see below); and (B) we proved that in general, the gap between them can be expressed in terms of the dimensions of the invariant measure. These relations are very general; they hold for all diffeomorphisms and flows on finite dimensional manifolds.
The results above are part of a subject called nonuniform hyperbolic theory. I would like to extend this theory to stochastic and infinite dimensional systems. Averaging effects of random noise should simplify the dynamical picture [2]. Extension to infinite dimensions will expand the scope of this theory to include semiflows defined by, e.g., dissipative parabolic PDEs; see [10] for a first step.
Natural invariant measures
For Hamiltonian systems, Liouville measure is clearly the natural invariant measure. What plays the role of Liouville measure for dissipative systems, such as those with attractors? The short answer is: SRB measures  except that things are a bit more complicated. In the 1970s Sinai, Ruelle and Bowen discovered these measures for uniformly hyperbolic attractors, a class of chaotic attractors satisfying strong geometric conditions. This body of ideas, minus the assertion of existence, was extended to general attractors by Ledrappier and myself (see [1]). Not every chaotic attractor admits an SRB measure, however, and it is very hard to determine if a given attractor does or not. These questions have remained unsettled; [3],[4] and [7] are among the few results known, and [3] was the first time SRB measures were constructed for genuinely nonuniformly hyperbolic attractors.
Decay of time correlations
By definition, deterministic dynamical systems have memory. The more chaotic a system is, the more rapidly it mixes up its phase space geometry, equivalently, the faster the decay of its time correlations (with respect to smooth test functions). The following two sets of results are my main contributions in this topic [4],[5]: (A) Via a socalled "tower" construction, I connected sufficiently hyperbolic (or chaotic) systems to countable state Markov chains. Leveraging these Markovlike structures, I showed that many statistical properties of the system are determined by tail properties of return times to certain reference sets. For example, exponentially decaying tails lead to exponential correlation decay, central limit theorems, large deviation principles, etc. (B) I proposed to connect the tail properties above directly to the geometry of the map, promoting the idea that to gain insight into the mode of correlation decay for a system that is predominantly hyperbolic, one should focus on its most nonhyperbolic parts. (A) and (B) together offered a unified way to get a handle on statistical properties of large classes of dynamical systems. I demonstrated that on a few examples, including the 2D periodic Lorentz gas [4]; others have used this method many more times.
Strange attractors
Even though there is no formal definition of a "strange attractor", everyone agrees that its dynamics cannot be simple. Wang and I undertook a systematic study of what can be thought of as "strange attractors of the simplest kind". We called them rankone attractors, referring to the fact that while these attractors can live in phase spaces of any dimension, they have only one direction of instability, with strong contraction in all other directions. One might expect to see such attractors following a regime's loss of stability, and that is what my coauthors and I (and others) have shown: rankone attractors occur naturally in periodically kicked oscillators, in periodically forced systems undergoing Hopf bifurcations, with homoclinic loops, and in certain slowfast systems. They can occur in systems defined by ODEs as well as PDEs; see e.g. [9].
This project consisted, in fact, of two separate parts, the second of which is reported above. The first part was a 130 page paper in which Wang and I identified a set of geometric conditions and proved that they imply the existence of rankone attractors [7]. This part of our work benefited from the techniques of Benedicks and Carleson, in their analysis of the Hénon maps. We extracted certain ideas from this one example, and developed them into a general class of dynamical systems that includes all of the examples above. We also gave a full description of the geometric and statistical properties of the attractors in this class.
Nonequilibrium dynamics
Systems in the real world do not operate in isolation; they are driven by external forces, and interact with the outside world. Much of current dynamical systems theory ignores such interactions, understandably so for reasons of simplicity. Below are two of my attempts to "push the envelope":
(A) An important source of inspiration is nonequilirbium statistical mechanics. In [6], Eckmann and I investigated the steady states of a class of mechanical chains connected to two unequal heat baths. Our aim was to elucidate how (a) dynamical properties and (b) local thermal equilibria factor in the determination of macroscopic observations such as mean energy and particle density profiles.
(B) Demers and I studied systems with holes: once the orbit of a point enters a "hole", it is lost forever. Starting from an initial distribution, relevant questions include the rate at which mass escapes, surviving distributions, etc. For a prototypical result, see [8], which treats a billiard table with holes.
SELECTED PUBLICATIONS
 (with F. Ledrappier) The metric entropy of diffeomorphisms, Part I: Characterization of measures satisfying Pesin's entropy formula, Part II: Relations between entropy, exponents and dimension, Annals of Math., 122, (1985), 509574.
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 (with F. Ledrappier) Entropy formula for random transformations, Prob. Th. Rel. Fields, 80, (1988), 217240.
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 (with M. Benedicks) SinaiBowenRuelle measures for certain Hénon maps, Invent. Math, 112, (1993), 541576.
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 Statistical properties of dynamical systems with some hyperbolicity, Annals of Math., (1998), 585650.
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 Recurrence times and rates of mixing, Israel J Math. 110, (1999), 153188
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 (with J.P. Eckmann) Nonequilibrium Energy Profiles for a Class of 1D Models, Commun. Math. Phys., 262, (2006), 237267.
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 (with Q.D. Wang) Toward a theory of rank one attractors, Annals of Math., 167, No.2, (2008), 349480.
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 (with M. Demers and P. Wright) Escape rates and physically relevant measures for billiards with small holes, Commun. Math. Phys., Vol 294, 2, (2010), 353338.
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 (with K. Lu and Q. D. Wang) Strange attractors for periodically forced parabolic equations, AMS Memoirs, (2013).
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 (with Z. Lian) Lyapunov exponents, periodic orbits and horseshoes for semiflows on Hilbert spaces, J. Amer. Math. Soc., 25 (2012), 637665.
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SELECTED REVIEW ARTICLES
 Ergodic theory of differentiable dynamical systems, "Real and Complex Dynamics", Ed. Branner and Hjorth, NATO ASI series, Kluwer Academic Publishers (1995), 293336.
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 Developments in chaotic dynamics, AMS Notices, Nov. 1998.
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 (with N. Chernov) Decay of correlations for Lorentz gases and hard balls, Encycl. of Math. Sc., Math. Phys. II, Vol. 101, Ed. Szasz, (2001), 89120.
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 What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 Issue 5, (2002) 733754.
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 (with K. Lin) Dynamics of periodicallykicked oscillators, J. Fixed Point Theory Appl., 7 no. 2 (2010), 291312.
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 Mathematical theory of Lyapunov exponents, J. Phys. A: Math Theor. 46 (2013) 254001.
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