### A Lie connection between Hamiltonian and Lagrangian optics.

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Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.

A quantum dynamical system, mimicking the classical phase doubling map $z\mapsto {z}^{2}$ on the unit circle, is formulated and its ergodic properties are studied. We prove that the quantum dynamical entropy equals the classical value log2 by using compact perturbations of the identity as operational partitions of unity.