This paper by Rufus Bowen explores the concept of topological entropy for uniformly continuous maps on metric spaces, particularly focusing on its relationship with measure-theoretic entropy. The topological entropy \( h_d(T) \) is defined for a map \( T \) on a metric space \( (X, d) \), and the paper provides general statements and calculations for this entropy in various contexts, including Lie groups and homogeneous spaces. Key results include: 1. **Equality of Entropies**: For a compact metrizable group \( G \) and a surjective endomorphism \( A \), the topological entropy \( h_{\mu}(R_g \circ A) \) equals the measure-theoretic entropy \( h_{\mu}(A) \) when \( \mu \) is the normalized right Haar measure. 2. **Linear Maps**: The topological entropy of a linear map \( T: R^m \to R^m \) is bounded by \( \max \{0, m \log \sup_{x \in M} \|dT\|_{\text{max}}\} \), and it is given by \( \sum_{|\lambda_i| > 1} \log |\lambda_i| \) where \( \lambda_i \) are the eigenvalues of \( T \). 3. **Homogeneous Spaces**: For a quotient space \( G / \Gamma \) where \( \Gamma \) is a uniform discrete subgroup of a locally compact metrizable group \( G \), the topological entropy of a map induced by \( T(x) = g A(x) \) is calculated, showing that \( h_d(T) = h_e(S) + h_d(\tau) \). The paper also discusses the relationship between topological entropy and measure-theoretic entropy, proving that for a continuous map \( T \) on a compact metric space, \( h(T) = h_d(T) \) when \( T \) preserves the Haar measure. Additionally, it provides formulas for the entropy of affine maps on Lie groups and homogeneous spaces, extending known results for tori.This paper by Rufus Bowen explores the concept of topological entropy for uniformly continuous maps on metric spaces, particularly focusing on its relationship with measure-theoretic entropy. The topological entropy \( h_d(T) \) is defined for a map \( T \) on a metric space \( (X, d) \), and the paper provides general statements and calculations for this entropy in various contexts, including Lie groups and homogeneous spaces. Key results include: 1. **Equality of Entropies**: For a compact metrizable group \( G \) and a surjective endomorphism \( A \), the topological entropy \( h_{\mu}(R_g \circ A) \) equals the measure-theoretic entropy \( h_{\mu}(A) \) when \( \mu \) is the normalized right Haar measure. 2. **Linear Maps**: The topological entropy of a linear map \( T: R^m \to R^m \) is bounded by \( \max \{0, m \log \sup_{x \in M} \|dT\|_{\text{max}}\} \), and it is given by \( \sum_{|\lambda_i| > 1} \log |\lambda_i| \) where \( \lambda_i \) are the eigenvalues of \( T \). 3. **Homogeneous Spaces**: For a quotient space \( G / \Gamma \) where \( \Gamma \) is a uniform discrete subgroup of a locally compact metrizable group \( G \), the topological entropy of a map induced by \( T(x) = g A(x) \) is calculated, showing that \( h_d(T) = h_e(S) + h_d(\tau) \). The paper also discusses the relationship between topological entropy and measure-theoretic entropy, proving that for a continuous map \( T \) on a compact metric space, \( h(T) = h_d(T) \) when \( T \) preserves the Haar measure. Additionally, it provides formulas for the entropy of affine maps on Lie groups and homogeneous spaces, extending known results for tori.