Infinite words can be fixed points of morphisms, and if the morphism is primitive, then such a word determines a unique dynamical system: the set of infinite words which have the property that each finite subword occurs in the fixed point word. The map on the dynamical system is the shift. Two dynamical systems are isomorphic if there exists a bi-continuous bijection between them which preserves the dynamics. We call two primitive morphisms dynamically equivalent if their dynamical systems are isomorphic. The task is to decide when two morphisms are dynamically equivalent. A morphism is called uniform if all the images of the letters have the same length. A first result is that the number of morphisms (of morphisms with the same length) dynamically equivalent to a given uniform morphism is finite, if the morphisms are one-to-one and if we ignore changes of alphabet. We will present the equivalence class of the Toeplitz morphism 0 → 01, 1 → 00. This is joint work with Ethan Coven and Mike Keane.
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