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An infinite permutation can be defined as a linear ordering of the set ofnatural numbers. In particular, an infinite permutation can be constructed withan aperiodic infinite word over $\{0,\ldots,q-1\}$ as the lexicographic orderof the shifts of the word. In this paper, we discuss the question if aninfinite permutation defined this way admits a canonical representative, thatis, can be defined by a sequence of numbers from [0, 1], such that thefrequency of its elements in any interval is equal to the length of thatinterval. We show that a canonical representative exists if and only if theword is uniquely ergodic, and that is why we use the term ergodic permutations.We also discuss ways to construct the canonical representative of a permutationdefined by a morphic word and generalize the construction of Makarov, 2009, forthe Thue-Morse permutation to a wider class of infinite words.

Canonical Representatives of Morphic Permutations