The Impact of Chaos on Science and Society (UNU, 1997, 415 pages) |

4. The impact of chaos on mathematics |

It is time to conclude my list of examples of encounters between chaos and mathematics. I have not tried to make the list complete, but I have tried to make it varied and diverse, showing that the relations between mathematics and the other sciences can take all kinds of different forms. My last example is that of dynamical zeta functions.

If *f*:*M*_{}

_{}

where Fix
*f*^{m}={*x*ÎM:*f*^{m}*x*=*x*}.
It often happens that *z*(*z*) converges and can be extended to a
meromorphic function in a more or less extended domain. The analytic properties
of *z*, and the position of the poles reflect subtle properties of the
map *f*. Following the example of the Riemann zeta function, dynamical zeta
functions have been introduced by Artin and Mazur [28] and Smale [5], and
generalized by myself. My own interest was induced by the possibility of using
methods of statistical mechanics to study these functions [29, 30]. Parry and
Pollicott [31] showed that a certain zeta function (related to the geodesic flow
on a manifold of negative curvation) has analyticity properties similar to those
of the Riemann zeta function, so that the proof of the prime number theorem can
be imitated. They obtained in this manner a "prime number theorem" describing
the distribution of lengths of closed geodesics on a compact manifold of (not
necessarily constant) negative curvature. This is a spectacular mathematical
result, but not directly related to chaos. As usual, however, mathematical
progress in the study of dynamical systems is associated with interesting
physical developments. Let us mention in this direction the important work of
Cvitanovic and collaborators [32] on zeta functions related to the Feigenbaum
operator.

Another development involving zeta functions is that of
*resonances*. If one takes the Fourier transform of a time autocorrelation
function one obtains a so-called *power spectrum*. In some cases at least
(Axiom A systems) one can prove that the power spectrum extends analytically
into the complex, with poles related to the poles of zeta functions. These poles
can be interpreted as dynamical resonances, and their study should help in
understanding the nature of chaos (see Pollicott [33], Ruelle [34], Baladi et
al. [35]).

At this time, our understanding of the analytic structure of zeta functions (and power spectra) is limited to rather special classes of dynamical systems. The mathematical interest of the subject stimulates a lot of research (see the recent work by Baladi and Keller [36], and Ruelle [37]), and one can hope that the results obtained will lead to further understanding of the physics of chaos.