Braid invariant for periodic points of surface maps and its applications

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We studied the topological structure of the set of periodic points for embeddings of a 2-dimensional disk to itself, by exploiting the braid invariant, which is one of the topological invariants defined for periodic points. Our results are the following :1. Thurston's theory has shown that if the embedding has a periodic point which is topologically complicated (i.e., its braid invariant has a pseudo-Anosov component in its decomposition), then there exist infinitely many periodic points. We proved that if P(n), the set of periodic points with period less than or equal to an integer n, is topologically complicated, then P(n) has at least 2n+3 points.2. When P(n) has at most 2n+2 points, the above result shows that this set is topologically simple. In this case, we determined all the possible forms of the braid invariant of P(n).3. We studied fixed points from a viewpoint different from the above, and obtained the following :(1) Using the notion of a braid, we introduced an equivalence relation on the fixed point set, and proved that the braid invariant of each equivalence class is of a simple type. This implies that the study of the structure of the fixed point set is divided into two parts: the study of the property of each equivalence class and that of how the equivalence classes are combined together.(2) We proved that the fixed point index of each equivalence class is less than two. As an application of this result, we studied the relationship between the topological property and stability of fixed points, and showed that every equivalence class with at least two points must contain an unstable fixed point. Moreover, we showed that the number of equivalence classes having an unstable fixed point is greater than that of the equivalence classes containing no unstable fixed points.