"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."
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