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close this bookThe Impact of Chaos on Science and Society (UNU, 1997, 415 pages)
close this folder4. The impact of chaos on mathematics
View the document(introduction...)
View the documentAbstract
View the documentI. Introduction: A historical view
View the documentII. The Lorenz attractor
View the documentIII. The Feigenbaum bifurcation
View the documentIV. Hydrodynamic turbulence
View the documentV. Ergodic theory of differentiable dynamical systems: Axiom A systems
View the documentVI. Ergodic theory of differentiable dynamical systems: General systems
View the documentVII. Quadratic maps of the interval and the Hénon attractor
View the documentVIII. Zeta functions
View the documentIX. Conclusion
View the documentReferences

VIII. Zeta functions

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

M is a map and j:M

C a function on M, define the formal power series

where Fix fm={xÎM:fmx=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.