Non-locally connected spaces I: Coverings and branched coverings on metric spaces

We give a definition of coverings and branched coverings on general (non-locally connected) metric spaces and study their lifting properties. Instead of lifting curves the theory is developed by lifting continua. An alternative homotopy theory is depicted and versions generalizing the classical results are given. We study applications to the dynamics of sphere branched coverings and provide several examples illustrating the interest of the results obtained.