**Dynamical Systems**

Dynamical Systems are an exciting branch of Mathematics.
They use results and methods from many other branches and on
the other hand find application in many branches of
mathematics and other sciences. They are also recognized as
important: two out of four Fields Medals (mathematical
equivalent of Nobel prize) at the last International
Mathematical Congress were awarded to researches in
dynamics: Artur
Avila and Maryam Mirzakhani, by the way the first woman
medalist in history. Four years earlier also two out of four
medals were awarded for the results based on Dynamical
Systems: Elon Lindenstrauss for applications of ergodic
theory to number theory and
Stanislav Smirnov for applications to statistical
physics.

The applications of
Dynamical Systems start from those to “most pure” of
mathematical disciplines - number theory. The relationship
is a long lasting one: from Chebyshev’s results at the
beginning of 20^{th} century through the works of
Furstenberg, Margulis,
McMullen and Tao (three last are Fields medalist as well) to the recent
results of Lindenstrauss mentioned above.

Then come disciplines considered more practical Differential
Equations and Probability Theory. The first is everyday
bread and butter of every engineer, the second of any
statistician or financial specialist. The Differential
Equations are inseparable from Dynamical Systems, for
serious research they use dynamical methods, the notions of
stability and all its variations are basic to Differential
Equations. Let
us mention the famous Lorenz strange attractor which was a
center of attention for the last 60 years.

Probability Theory uses many measure theoretic results which
were first proven in Dynamical Systems. The most basic
theorem of probability the Strong Law of Large Number is
nothing else but Birkhoff Ergodic Theorem , also fundamental
result of Dynamical Systems. Kolmogorov, the creator of
mathematical Probability Theory was also one of the great
figures of Dynamical Systems.

Let us go back to more abstract applications. One of the
most famous mathematical objects is the Mandelbrot set, a
beautiful fractal which is seen in innumerable images and
represents a set studied in the dynamics of quadratic
polynomials. Fractals have also practical applications, for
example in material science. If the brakes of your car work
properly it is also because the pads maker used methods
based on fractal theory of surfaces. Another surprising
application, every time you connect by your cell phone the
connection is possible due to DS-CDMA algorithm
based on a chaotic map of an interval, a toy model of
every Dynamical Systems enthusiast.

Dynamical Systems methods are used also as theoretical
tools in other sciences. In Statistical Physics the border
between Mathematics and Physics became more ideological
than real. Most recent results in Statistical Physics are
actually results in Dynamical Systems, obtained both by
mathematicians and physicists,
often working together. A new discipline, Mathematical
Physics, was created and is blossoming. In Biology
mathematical (Dynamical Systems) models successfully
describe work of the heart and the propagation of
diseases. In Chemistry the famous Belousov-Zhabotinsky
reaction is an example of a periodic behavior and has
numerous Dynamical Systems models.