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Ian Morris
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Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform frequencies and recurrent properties. where $p_{ij}$ is the frequency of block $(ij)$ in the sequence. As we all known, uniform frequencies equivalent to unique ergodicity in dynamical systems, a good example is Sturmian sequence which has uniform frequencies and recurrent properties, but the frequencies are determined by the irrational rotation.