Texas A&M University, Department of
Mathematics, 216 Milner Hall, 9th of February 2005, 3:00-3:50
Groups and Dynamics Seminar
(joint session with Mathematical
Physics and Harmonic Analysis Seminar)
Symbolic dynamics
for the modular surface
Svetlana Katok of Pennsylvania State
University
There are two essentially different methods of coding geodesics on
surfaces of constant negative curvature. A geometric method stems
from a 1898 work of Hadamard and was developed by Morse and
Hedlund in the 1920's and 30's. It consists of recording the successive
sides of a given fundamental region cut by the geodesic, and may
be applied to all finitely generated Fuchsian groups.
For the modular surface with the standard fundamental region it
produces a bi-infinite sequence of non-zero integers which records
oriented excursions of the geodesic into the cusp.
Another method, which we call arithmetic, is specific for the
modular surface and uses continued fraction expansions of the end
points of the geodesic at infinity and so-called "reduction theory".
This method of study and classification of integral binary quadratic
forms goes back to a 1801 work of Gauss. In the second half of
the XIX century Dirichlet described Gauss's reduction algorithm using
simple continued fractions, and Markov and Hurwitz extended
it to classify binary quadratic forms with real coefficients. This
method was introduced to dynamics by a 1924 paper of Artin, where
he used it to exhibit dense geodesics on the modular surface.
For 80 years these classical works provided inspiration for
mathematicians and a testing ground for new methods in dynamics,
geometry and combinatorial group theory. Major contributions were made
by R. Bowen, C. Series, R. Adler and L. Flatto, D. Grabiner and
J. Lagarias, P. Arnoux, who interpreted the classical
works in the modern language of symbolic dynamics and expanded
the field.
Quite surprisingly, there was still room left for new results in
this well-developed area.
We will present three arithmetic methods for coding oriented
geodesics on the modular surface using various continued fraction
expansions and corresponding reduction theories and show that the
space of admissible coding sequences for each code is a one-step
topological Markov chain on a countable alphabet, or sometimes a
Bernoulli system. In contrast with arithmetic codes, the set of
admissible geometric coding sequences is quite complicated. We will
identify a maximal one-step topological Markov chain of
admissible geometric codes, and other classes of admissible geometric
codes for which geometric and arithmetic codes coincide and which are
mysteriously related to the five Platonic solids.