Insights into loss of ergodicity by symmetry breaking in expanding systems of coupled maps
Bastien Fernandez
Abstract.
The question has been open for a long time to exhibit, and prove, purely
deterministic particle systems with Ising-type phenomenology, namely chaotic
systems of interacting units whose attractor looses ergodicity via symmetry
breaking as the strength of interactions increases.
In this talk, the dynamics of a suitably chosen family of N coupled expanding
circle maps will be investigated in a parameter regime where absolutely
continuous invariant measures are known to exist. Empirical evidence will be
given of symmetry breaking of the ergodic components upon increase of the
coupling strength, suggesting that breaking of ergodicity should occur for
every integer N > 2. Computer assisted-proof approaches will be discussed and
evaluated, which aim to rigorously construct, for arbitrary N, asymmetric
ergodic components of positive Lebesgue measure.
Due to the explosive growth of the required computational resources, these
approaches have been completed for small values of N only. However, they
suggest that Ising-type phenomenology should be provable for systems of
arbitrary number of particles with erratic dynamics, in a purely deterministic
setting, without any reference to random processes.