Miaohua Jiang
SRB Measures for Lattice Dynamical Systems
(419K, pdf)

ABSTRACT.  For weakly coupled expanding maps 
on the unit circle, Bricmont and Kupiainen showed that 
the Sinai-Ruelle-Bowen (SRB) measure exists as a Gibbs state. 
Via thermodynamic formalism, we prove that 
this SRB measure is indeed the unique equilibrium state for a H\"older continuous 
potential function on the infinite dimensional phase space. 
For a more general class of lattice systems that are small perturbations 
of the uncoupled map lattice, we present the variational principle, the entropy formula, 
and the formula for the potential function for the SRB measures. For coupled 
map lattices with nearest neighbor interactions, we give an explicit formula 
of the potential function for the SRB measure and consequently, obtain 
the entropy in terms of coupling parameters.