| Summary: | 碩士 === 國立臺灣大學 === 數學研究所 === 97 === A continuous map f:[0,1]->[0,1] is called an n-modal map if there is a partition P={0=z_0<z_1<...<z_n=1} such that f(z_2i)=0, f(z_2i+1)=1 and, f is monotone on each [z_i,z_i+1]. It was proved by Milnor and Thurston (1977) that there exists a topological semi-conjugacy from a piecewise strictly monotone map to a piecewise linear map. In this article, we give a method for constructing the topological semi-conjugacy numerically which extends the results from Fotiades, Boudourides (2001) and Banks, Dragan, Jones (2003). In addition, the uniqueness of the semi-conjugacy, is proved by this method. The convergence rate is discussed for the approximation method also. Moreover, in contrast to Fotiades and Banks who only consider condition which ensure the conjugacy map exists, here we state equivalent conditions for the semi-conjugacy to be exactly a bijection, which coincide with Parry''s (1966) result. Finally, two applications are given. In one, we study the trajectory of the invariant Cantor set for the logistic map l_mu(x)=mu x(1-x) when the parameter mu>=4. In the other, we construct an invariant measure for an n-modal map.
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