Consider dynamical systems $(X,T)$ where $X$ is a compact metric space, $T:X\rightarrow X$ is continuous, the system is minimal and finally, $0<h\_{\rm{top}}(X)<\infty$. I am looking for examples of such systems that do not admit a measure of maximal entropy (mme). Non-minimal topological systems without mme are easy to construct, so I am asking for examples of minimal systems without mme. 



Examples of minimal topological systems which are not intrinsically ergodic
The question 
[Transitive shifts with multiple fully supported MMEs](https://mathoverflow.net/questions/43564/transitive-shifts-with-multiple-fully-supported-mmes) 
that @DanRust mentioned above also discusses MME's, but it is concerned with a different way that a system may fail to be intrinsically ergodic, i.e. having multiple MME's. Grillenberger's name comes up in that post. After some search I found this [article](https://link.springer.com/content/pdf/10.1007/BF01540599.pdf) of his where he constructs examples I was looking for as certain shift systems. 


