"The zeta function, non-differentiability of pressure, and the critical exponent of transition"

A.O. Lopes

(from Advances in Mathematics 101 (1993) 133-165)


[Abstract:] "The main purpose of this paper is to analyze the lack of differentiablility of the pressure and, from the behaviour of the pressure around the point of non-differentiablity, to derive an asymptotic formula for the number of periodic orbits (under certain restrictions related to the norm of the periodic orbit) of a dynamical system. This kind of result is analogous to the well known Theorem of Distribution of Primes of Introduction to Analytic Number Theory (T.M. Apostol, 1976, Springer-Verlag, New York/Berlin). This result follows from analysis of the dynamic zeta function and Tauberian theorems. We introduce a functional equation relating the pressure and the Riemann zeta function, and this equation plays an essential role in the proof of our results. We can say, in general terms, that the result presented here extends some well known results obtained for expanding dynamical systems to a certain class of non-expanding dynamical systems. From another point of view, we can say that we are analyzing thermodynamic formalism for non-Holder functions (or for functions not in the class Ftheta considered by D. Ruelle, W. Parry, and M. Pollicott). As an example of the results presented here, we show that for the Manneville-Pomeau map f:[0,1] -> [0,1], given by f(x) = x + x1+s (mod 1), where s is a positive real constant, 0.5 < s < 1, the pressure

p(t) = supv in M(f) {h(v) - t Integral log|f'(x)|dv(x)}
is such that

p(t) ~= h(mu)(1 - t) + B(1 - t)1/s, for t < 1
and ~= 0 for t greater than or equal to 1,

where B is a constant and h(mu) is the entropy of the Bowen-Ruelle-Sinai measure. The above result is an example of a first order phase transition. In this case, the pressure is not differentiable at t = 1, and s-1 will be the critical exponent of transition. Some of the results can also be seen as a result in number theory for partitions with weights. We give a proof of a result of B. Felderhof and M. Fisher (1970, Ann. Phys. (N.Y.) 58, 176-281; 1967, Physica 3, 255-283) concerning the critical exponent of transition in the setting of thermodynamic formalism."

 
archive      tutorial      mystery      new      search      home      contact