Abstract
A model involving a chain of N ≥ 2 spins s i = ±1, i = 1,…, N, evolvi ng syncronously in discrete time t via a nonlinear, autonomous transformation s i ( t+1) = s i ( t) s i+1 ( t), i = 1,…, N−1; s N ( t+1) = s N ( t), is presented. The transformation equations are solved explicitly and the detailed decomposition of state space into ergodic sets is found. On the assumption of equally likely initial states, the mean recurrence time is calculated and its variance is discussed. The model displays a strikingly sensitive dependence on the number of spins, and this is reflected in the “staircase” behavior of the mean recurrence time. Remarks are made regarding the connection between the behavior of the model and the ground states of a related two-dimensional Ising model.
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