Path connectedness of the space of hyperbolic ergodic measures

In 1977 Sigmund proved that the space of ergodic measures supported on a basic set of an Axiom A diffeomorphism is path connected. In the talk I will describe a substantial generalization of this result to the space of hyperbolic ergodic measures supported on an isolated homoclinic class of a general diffeomorphism. Such homoclinic classes should be viewed as basic structural elements of any dynamics. Examples will be discussed. This is a joint work with A. Gorodetsky.