Universite de Bourgogne Acknowledgements

First I want to thank my parents who taught me the love and courage of life. Without these, I would never be able to study my doctorate in mathematics and finish this thesis. Both of Lan Wen and Christian Bonatti, my advisors, are significant and essential to my study of dynamical systems during these years. I still remember the scene of Lan Wen taught me dynamical systems from zero in the first year of my doctorate, which opening the gate of my research in this field. I want to thank his patience and encouragements, which is so crucial and helpful to me, both in mathematics and daily life. To Christian, I seriously don’t know how to express my thanks and gratitude in language. Without him, this thesis would be definitely mission impossible. I could never even try to list all what I learned from him, mathematics, the French culture, music, painting, even French cheese. I want to thank Shaobo Gan. He is an example for me, not only as a mathematical researcher, but also an honest and decent person. I greatly appreciate Amie Wilkinson and Sylvain Crovisier for being the reporter of this thesis, which is an honor for me. This thesis is based on a lot of suggestions and comments from them. I want to thank Bin Yu who taught me a lot of knowledge of topology. The time we cooked together in France are wonderful memory for me. I owe Sébastien Alvarez a lot for helping me during my staying in Dijon, especially for translating the introduction of this thesis into French. I want to thank Rafael Potrie, Dawei Yang, Baolin He, Katsutoshi Shinohara, Doris Bohnet, Ana Rechtman, my classmates and friends for a lot discussion in mathematics. This work is mainly finished during my stay in IMB, Dijon, which is supported by the Chinese Scholarship Council. I want to thank them and the Chinese Embassy of France. I als