Are there examples of subshifts (that is, closed shift-invariant subsets of the full shift {$1...n$}${}^{\mathbb{Z}}$) on which the shift is topologically mixing, which admit a shift-invariant probability measure of full support, but no invariant ergodic measure of full support ?


I guess the answer is no, but I can't locate a reference. 


Also I would be interested in a topologically mixing subshift that has no invariant probability measure of full support.



topologically mixing subshifts without ergodic measures
A construction of a topologically transitive mixing subshift with a fully supported invariant measure, but no fully supported ergodic measure, is given by Benjamin Weiss in the article "Topological transitivity and ergodic measures", *Mathematical Systems Theory* 1971. It gives a direct combinatorial construction, with $n=2$.


It is easy to give an example of a topologically transitive subshift on two symbols with no fully supported measure: just take the orbit closure under the shift of the sequence with all negative entries equal to one, and all non-negative entries equal to two.


