My research interests are in the theory of dynamical systems
and and its connections with other branches of mathematics such as number
theory, geometry and representation theory. By a dynamical system I mean a
space equipped with a group of transformations. The orbit structure of these
transformations is typically very complicated: there are
orbits that fill the space densely or accumulate on an exotic fractal set.
Nonetheless, one can develop techniques to analyse statistical properties of orbits
over long time periods. This problem is not only of fundamental importance
in the theory of dynamical systems, but also has striking applications to number
theory. For instance, in the theory of Diophantine approximation one is
interested in the best possible approximation for a given real number
* x* by rational numbers. It
turns out that this problem is intimately related to dynamical properties
of the rotation of the circle

Another way to appoach this problem is to consider the action of the group SL(2,**Z**) on the real line by fractional linear transformations:

In both cases information about structure of orbits provides deep insights into Diophantine properties of real numbers.

**Distribution of
Orbits and Ergodic Theorems
**

One of the fundamental problems in ergodic theory is to
describe the asymptotic distribution of orbits. Let us consider a one-parameter
family of transformations **T _{t}** of a
space

** **

Roughly speaking, this average represents the "probability"
that **T _{t}**

Proving analogues of ergodic theorems for more general dynamical systems is an active area of research. See:

- V. Bergelson and A. Gorodnik, Trieste lectures on ergodic theorems and applications, lecture notes.
- A. Nevo, Pointwise ergodic theorems for actions of groups. Handbook of dynamical systems. Vol. 1B, 871--982, Elsevier B. V., Amsterdam, 2006.
- A. Gorodnik and A. Nevo, The ergodic theory of lattice subgroups, Annals of Mathematics Studies 172, Princeton University Press, 2010.
- A. Gorodnik and A. Nevo, Quantitative ergodic theorems and their number-theoretic applications.

Ergodic theorems describe behaviour of generic orbits, but for many applications it is crucial to establish asymptotic distribution of a given orbit. Remarkable results in this direction has been established by Dani, Margulis, and Ratner, see:

- D. Witte Morris, Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005.

These results lead to many far-reaching applications to number theory.

More generally, one is interested to understand finer statistical properties of
the sequence of samples
**f(T _{t}x
)**
computed along orbits.
For chaotic dynamical systems, one might expect that these samples are "quasi-independent" in a suitable sense.
This lead to analysis of correlations and higher-order correlations:

- D. Dologopyat, Limit Theorems for Hyperbolic Systems.
- A. Gorodnik, Higher-order correlations for group actions. to appear in Tata Institute Conference Proceedings.

For instance, one can show that
the samples
**f(T _{t}x
)** satisfy the Central Limit Theorem and
other probabilistic limit laws, as sequences of independent random variables.

**Dynamical Systems and Diophantine Geometry**

Quite a few problems in number theory can be modelled using methods for the theory of dynamical systems. The picture on the left represents the set of rational points on the projective plane. What is the distribution of these points? More generally, given a system of polynomial equations, what can one say about the set of rational solutions? Although this is one of the oldest problems in mathematics, it is still out of reach in its full generality, but when the system equations has a large group of symmetries, one may be able to analyse the set of solutions using dynamical techniques.

One of the most remarkable applications of dynamical systems to number theory was discovered by G. Margulis. It was conjectured by Oppenheim in late 20's that in particular the following set of real numbers is dense:

This is indeed the case, but it turns out to be surprisingly
difficult to prove. In fact the Oppenheim conjecture was proved by G. Margulis
only in 1987, and his proof uses dynamics on the space SL(3,**R**)/SL(3,**Z**).

There are many more surprising applications of dynamical systems to number theory. If you would like to know more, you may consult lecture notes from my courses:

- Flows on homogeneous spaces and applications (Bristol, Spring 2009),
- Diophantine approximation and Dynamics (TIFR, Mumbai, India, Summer 2010),
- Dynamical Systems in Number Theory (Bristol, Autumn 2010),
- Applications of Ergodic Theory of nonamenable groups to Number Theory (Tours, France, Spring 2011),
- Kunze-Stein phenomenon, ergodic theorems and applications (Lausanne, Switzerland, Spring 2011),
- Number Theory and Dynamical Systems (Vienna, Austria, Autumn 2011),
- Ergodic theory of large groups and number theory (Eidgen�ssische Technische Hochschule, Z�rich, Autumn 2012),
- Diophantine approximation and flows in homogeneous spaces (Institut Fourier, Grenoble, France),

- G. Margulis, Diophantine approximation, lattices and flows on homogeneous spaces. A panorama of number theory or the view from Baker's garden (Zurich, 1999), 280-310, Cambridge Univ. Press, Cambridge, 2002.
- E. Lindenstrauss, Some examples how to use measure classification in number theory. Equidistribution in number theory (A. Granville and Z. Rudnick, eds.), 261-303, Springer, 2007.
- D. Kleinbock, Metric Diophantine approximation and dynamical systems, lecture notes.
- M. Babillot, Points entiers et groupes discrets: de l'analyse aux systemes dynamiques, in Rigidite, Groupe Fondamental Et Dynamique, Societe Mathematique de France, 2002.
- A. Gorodnik and A. Nevo, Quantitative ergodic theorems and their number-theoretic applications.

**Rigidity Phenomena in Dynamical Systems**

Another direction of my research is various rigidity
properties of groups and group actions. Consider, for instance, the problem of
classification of isomorphisms of the group SL(n,**Z**). While the group
SL(2,**Z**) has a large number of distinct isomorphisms, the situation is
very different for n>2. It turns out that when n>2 every isomorphism of
SL(n,**Z**) is the restriction of a algebraic isomorphism of the ambient
group SL(n,**R**). This remarkable rigidity phenomenon was proved by G.
Prasad, following previous work of G. Mostow (see
Mostow rigidity).

An ambitious programme, proposed by Zimmer in 80's, is aimed
at classification of actions of groups like SL(n,**Z**), n>2, on compact
manifolds. It is believed that every such action is "built" from well-known
algebraic actions of the group. More generally given a dynamical system one may
ask, for instance, to describe

- maps that commute with the action,
- perturbations of the action,
- closed invariant sets,
- invariant measures, etc.

This programme has been actively pursued for several classes of actions. For further readings, see my courses

- Introduction to arithmetic groups (Bristol, Winter 2010),
- Lie groups, algebraic and arithmetic groups (Bedlewo, Poland, Summer 2011),
- Lattices in Lie groups (Institut de Math�matiques de Jussieu, Paris, France),

- R. Spatzier, An invitation to rigidity theory, Modern Dynamical Systems and Applications (ed. M. Brin, B. Hasselblatt, Y. Pesin), p. 211-231, Cambridge University Press 2004.
- G. Margulis, Problems and conjectures in rigidity theory. Mathematics: frontiers and perspectives, 161-174, Amer. Math. Soc., Providence, RI, 2000.
- M. Gromov and P. Pansu, Rigidity of lattices: an introduction. Geometric topology: recent developments (Montecatini Terme, 1990), 39-137, Lecture Notes in Math., 1504, Springer, Berlin, 1991.
- D. Fisher, Groups acting on manifolds: around the Zimmer program, preprint.