Breaking of Ergodicity in Expanding Systems of Globally Coupled
Piecewise Affine Circle Maps
To identify and to explain coupling-induced phase transitions in Coupled
Map Lattices (CML) has been a lingering enigma for about two decades. In
numerical simulations, this phenomenon has always been observed preceded by
a lowering of the Lyapunov dimension, suggesting that the transition might
require changes of linear stability. Yet, recent proofs of co-existence of
several phases in specially designed models work in the expanding regime
where all Lyapunov exponents remain positive.
In this talk, I will consider a family of CML composed by piecewise
expanding individual map, global interaction and finite number N of sites,
in the weak coupling regime where the CML is uniformly expanding.
I will show, mathematically for N=3 and numerically for N>3, that a
transition in the asymptotic dynamics occurs as the coupling strength
increases. The transition breaks the (Milnor) attractor into several
chaotic pieces of positive Lebesgue measure, with distinct empiric
averages. It goes along with various symmetry breaking, quantified by means
of magnetization-type characteristics.
Despite that it only addresses finite-dimensional systems, to some extent,
this result reconciles the previous ones as it shows that loss of
ergodicity/symmetry breaking can occur in basic CML, independently of any
decay in the Lyapunov dimension.