Unstable entropy in smooth ergodic theory AT is supported by FAPESP 2017/06463-3 and CNPq (PQ) 303025/2015-8. This work was finalized when the author visited Peking University and Université Paris-Sud (financial support: project ANR-16-CE40-0013 ISDEEC). The conversations with Sylvain Crovisier, Jiagang Yang, Mauricio Poletti and Jérôme Buzzi, were very useful for improving the results and I thank them. I would like to thank Yi Shi, Jinhua Zhang and the whole dynamical system group in Beijing an