Explore the intricacies of interval map dynamics in this comprehensive lecture on monotonicity of entropy, transfer operators, and holomorphic motions. Delve into the Ruelle-Thurston transfer operator and examine an explicit example involving disconnected quadratic Julia sets and eigenvalue limit distributions. Investigate applications to rational dynamics and compare Tsujii's and Milnor-Thurston's approaches to entropy monotonicity in the real quadratic family. Analyze a local approach using holomorphic motions, focusing on the transfer operator and its spectrum. Discover the main theorem and its applications, and consider the critically infinite case, questioning whether saddle-nodes unfold in a positive direction.