Krylov-Bogolybov theorem states that if $X$ is metrizable compact space and $f: X \rightarrow X$ is continous then it admits an invariant Borel probability measure.
I would like to build a map $F$ from $3$-dimensional ball to itself so that $F$ does not admit an invariant measure (by invariant measure we denote $\lambda$ so that $\lambda(A) = \lambda(F^{-1}A)$ for any $A$). Are there any common contructions to do this?
For example, it's reasonable to require discontinuity from $F$, whereas this doesn't immediately give a result.
Any help would be much appreciated
I have an idea but I am not sure it leads to the answer. If a map has an invariant measure, this measure can always be decomposed into ergodic measures. So it is enough to construct a map without ergodic measures. Now you can construct a map from the shift space to itself (note that any compact is measure-theoretically isomorphic to a shift on the space sequences of zeroes and ones with some, so later you can just take a composition with this isomorphism). To do it, you can just choose a favourite way of non-convergence and define the map in that way for every sequence independently so that the ergodic average does not everywhere.
What I am not sure about here -- formally one can not use measure-theoretic isomorphism, as it requires an invariant measure as a given. So if one constructs a bad map on the space of sequences and then using it define a map on you ball, it is not clear that the map on the ball will be bad as well. I guess one should take a closer look at what happen with a convergence of the averages of a continuous observable (necessary for ergodicity) after applying a homeomorphism.
But probably an example could be found in a book on abstract ergodic theory.