Orbit equivalent substitution dynamical systems and complexity

Abstract:

For any primitive proper substitution $\sigma$, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems $\{(X_{\zeta _n}, T_{\zeta _n})\}_{n=1}^{\infty }$ such that they all are (strong) orbit equivalent to $(X_{\sigma }, T_{\sigma })$. We show that the complexity of the substitution dynamical systems $\{(X_{\zeta _n}, T_{\zeta _n})\}$ is the essential difference that prevents them from being isomorphic. Given a primitive (not necessarily proper) substitution $\tau$, we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to $(X_{\tau }, T_{\tau })$.