The Ellis semigroup of a discrete dynamical system (X, f) is denoted by E(X, f). We only consider surjective continuous functions \(f:[0,1]\rightarrow [0,1]\) such that \(\{ f^n : n\in {\mathbb {N}}\}\) is infinite. The main result asserts that the Ellis semigroup E([0, 1], f) is always a compactification of the natural numbers with the discrete topology. We also prove some results estimating the cardinality of the Ellis semigroup of certain dynamical systems: we show that if \(f:[0,1]\rightarrow [0,1]\) is a continuous function with positive topological entropy, then \(|E([0,1],f)|= 2^{2^{\aleph _0}}\), and if \(f:[0,1]\rightarrow [0,1]\) is a \(2^{n}\)-function for some \(n \in {\mathbb {N}}\), then \(|E([0,1],f)-\{f^n:n\in {\mathbb {N}}\}|=2^n.\)

离散动力系统 ( X ,  f )的 Ellis 半群用E ( X ,  f ) 表示。我们只考虑满射连续函数\(f:[0,1]\rightarrow [0,1]\)使得\(\{ f^n : n\in {\mathbb {N}}\}\)是无限的. 主要结果断言 Ellis 半群E ([0, 1],  f ) 总是具有离散拓扑的自然数的紧化。我们还证明了一些估计某些动力系统的 Ellis 半群的基数的结果：我们证明如果\(f:[0,1]\rightarrow [0,1]\)是一个具有正拓扑熵的连续函数，那么\(|E([0,1],f)|= 2^{2^{\aleph _0}}\)，如果\(f:[0,1]\rightarrow [0,1]\)是一个\(2^{n}\) - 一些\(n \in {\mathbb {N} }\) 的函数，然后\(|E([0,1],f)-\{f^n: n\in {\mathbb {N}}\}|=2^n.\)