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The main theme of my research is a combinatorial study of complex dynamical systems in dimension two, namely the complex Henon maps. I am now working on a criterion of their hyperbolicity, topology and combinatorics of their Julia sets and of their parameter loci, and applications of such complex methods to real dynamics. On the other hand, I have established a "kneading theory" for Lozi maps of the plane as well as a "lap number formula" for their topological entropy. This enables us to compute the entropy of Lozi maps with rigorous error bounds. With physicists I am working on the relationship between quantum tunneling phenomena and complex dynamics. In particular, I have formulated and solved a conjecture suggested by them.