We prove some results concerning the entropy of Darboux (and almost continuous) functions. We first generalize some theorems valid for continuous functions, and then we study properties which are specific to Darboux functions. Finally, we give theorems on approximating almost continuous functions by functions with infinite entropy.
Given a vector field X on an algebraic variety V over ℂ, I compare the following two objects: (i) the envelope of X, the smallest algebraic pseudogroup over V whose Lie algebra contains X, and (ii) the Galois pseudogroup of the foliation defined by the vector field X + d/dt (restricted to one fibre t = constant). I show that either they are equal, or the second has codimension one in the first.
Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure . We consider the map defined on X × G by and the cocycle generated by φ. Using a characterization of the ergodic invariant measures for , we give the form of the ergodic decomposition of or more generally of the -invariant measures , where is χ∘φ-conformal for an exponential χ on G.
For several specific mappings we show their chaotic behaviour by detecting the existence of their transversal homoclinic points. Our approach has an analytical feature based on the method of Lyapunov-Schmidt.