This paper constructs \(K3\) surface automorphisms with small entropy. Let \(F : X \rightarrow X\) be an automorphism of a compact complex surface \(X\). The topological entropy \(h(F )\) is determined by the spectral radius of \(F^*\) acting on \(H^\bullet (X)\). Indeed, \(h(F ) = \log p(F^* |H^2 (X))\). If \(h(F ) > 0\), then a minimal model for \(X\) is either a \(K3\) surface, an Enriques surface, a complex torus, or a rational surface. The lower bound \(h(F ) \geq \log\lambda_{10}\) holds for all surface automorphisms of positive entropy. Here \(\lambda_{10}\) is Lehmer's number, which is the smallest known Salem number. (A Salem number \(\lambda > 1\) is an algebraic integer which is conjugate to \(1/\lambda\), and whose remaining conjugates lie on \(S^1\). There is a unique minimum Salem number \(\lambda_d\) of degree \(d\) for each even \(d\).)The main results of the paper under review are formulated as follows. Theorem 1. There exists an automorphism of a non-algebraic \(K3\) surface with entropy \(h(F ) = \log\lambda_{10}\). \(K3\) surfaces in Theorem 1 are not projective. The next result is concerned with projective \(K3\) surfaces. Theorem 2. There exists an automorphism of a projective \(K3\) surface with entropy \(h(F ) = \log\lambda_6\). Theorem 3. There exists an automorphism of a complex torus \(\mathbb{C}^2 /\Lambda\) with entropy \(h(F ) = \log\lambda_6\), and an automorphism of an abelian surface with entropy \(h(F ) = \log\lambda_4\). In each case, no smaller positive entropy is possible. To construct these examples, first the general theory is developed for equivariant gluing, Coxeter groups and twists. They are then used to produce a model of the desired lattice automorphism \(F^* : H^2 (X, \mathbb{Z}) \rightarrow H^2 (X, \mathbb{Z})\). Finally the lattice automorphisms are related to automorphisms of \(K3\) surfaces (via the strong Torelli theorem and surjectivity of the period map). For instance, to exhibit an automorphism \(F : X \rightarrow X\) of a non-algebraic \(K3\) surface \(X\) with minimum entropy \(h(F ) = \log\lambda_{10}\), first construct a lattice automorphism \(f : L \rightarrow L\) with characteristic polynomial \(\det(xI - f ) = P_{10} (x)(x - 1)^9 (x + 1)(x^2 + 1)\) satisfying certain condition. Here \(P_{10} (x)\) is the minimal polynomial of \(\lambda_{10}\). The pair \((L, f )\) is obtained by gluing \((L_1 , f_1 )\) and \((L_2 , f_2 )\), where the constituents correspond to the specific factorization of the characteristic polynomial \(((L_1 , f_1 )\) to \(P_{10} (Xx)\) and \((L_2 , f_2 )\) to the rest). The results are obtained by detailed study of each constituent.