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Hi Vaughn,

It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology. The proof is very neat and takes only one a page or so. Here is the paper:

Sigmund, Karl "On the connectedness of ergodic systems. systems." Manuscripta Math. 22 (1977), no. 1, 27–32.

I don't know about generalizations. Sigmund's proof does not generalize directly.

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Hi Vaughn,

It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology. The proof is very neat and takes only one page or so. Here is the paper:

Sigmund, Karl On the connectedness of ergodic systems. Manuscripta Math. 22 (1977), no. 1, 27–32.

I don't know about generalizations. Sigmund's proof does not generalize directly.