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 transitivemixing, 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 transitive mixing subshift that has no invariant probability measure of full support.

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 transitive, 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 transitive subshift that has no invariant probability measure of full support.