一維函數之共軛性

Abstract

若一連續函數 f:[0,1]-&gt[0,1] ，存在一組分割 0=z_0&ltz_1&lt...&ltz_n=1 使 f(z_2i)=0 、 f(z_2i+1)=1 且 f 在 [z_i,z_i+1] 區間為單調函數，則稱 f 為 n-modal。Milnor 及 Thurston (1977) 最先給出了一個片段絕對單調函數至片段線性函數之 semi-conjugacy 的存在性。本篇論文為推廣 Fotiades, Boudourides (2001) 及 Banks, Dragan, Jones (2003) 的方法，建構 n-modal 函數到 tent map 之 semi-conjugacy ，並更進一步利用此方法證出 semi-conjugacy 的唯一性。此方法可用於數值計算 n-modal 映射之 semi-conjugacy ，並詳細估計出其收歛性。由於前述 Fotiades 及 Banks 等人只給了當 conjugacy 存在的結果，本文給出所構造出的 semi-conjugacy 為一對一映成函數之等價條件，這些條件驗證了 Parry (1966) 的結果。本文最後給了兩個應用：一個是研究 logistic map l_mu(x)=mu x(1-x) 之 invariant Cantor set 隨 mu&gt=4 變化之軌跡，另一個是可建構 n-modal map 之保測變換。

A continuous map f:[0,1]-&gt[0,1] is called an n-modal map if there is a partition P={0=z_0&ltz_1&lt...&ltz_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&gt=4. In the other, we construct an invariant measure for an n-modal map.