The study of open maps is an exciting relatively new area of the theory of dynamical systems. The characterisation of the holes involves geometry (‘shape’) and measure theory (‘size’), so their study involves a complex interplay between dynamics, geometry and analysis.
The standard dynamical approach is to assume that the survivor set (= the points whose orbits do not fall into a hole) is “sufficiently large” and investigate ergodic and geometric properties of the induced map. The novelty of the proposed project is to take a step back and give sufficient conditions that the survivor set is indeed "sufficiently large" (uncountable, say) for the induced map to be meaningful. This involves a detailed analysis of the class of holes under investigation.
The classes of maps under investigation include - but are not limited to - expanding maps of the interval, algebraic toral automorphisms and subshifts.