| We study a new class of maps, Continuous Baker maps, which are an extension of Pseudo-Anosov diffeomorphisms. These maps are hyperbolic in an open and dense set with a finite number of both periodic and non-periodic orbits of singularities. In addition, they may contain a finite number of bristles, which are points with a countable number of stable and unstable separatrices. We describe how to create these maps and determine their various topological and geometric properties.;We conclude that Continuous Baker maps have nonzero topological entropy and have the smallest topological entropy in their homotopy class, relative to the closure of the singularities. The Lefschetz index can be calculated for all periodic points of Continuous Baker maps, whether regular, singular, or bristle points. This index is used to prove that Continuous Baker maps have the fewest number of fixed points in their homotopy class, relative to the closure of the singularities. The Gauss-Bonnet theorem can be extended to give a result relating the bristle points and singularities of the foliations with the Euler characteristic of the surface. |
