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.