Boshernitzan, in the 1980s, proved a bound on the number of ergodic measures for minimal subshifts of low complexity. Cyr-Kra (2019) extended this work by1) removing the minimality requirement and
2) proving the bound for the number of generic measures.
In this talk, I will describe the background and sketch Cyr-Kra's proof. 
Reference : Van Cyr, Bryna Kra, Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. 21 (2019), no. 2, pp. 355–380