Measure Preserving Homeomorphisms at Fixed Points

In an article of a few years ago [2] Kerékjartó obtained interesting results about certain types of transformations which he called similitudes. With a few modifications and extensions his methods can be used to gain information about the structure of measure preserving transformations at fixed points. For simplicity the results are formulated for Euclidean w-space although they could easily be given a much more general setting and in particular the relevant ones apply to any ^-dimensional manifold on which there is defined a measure satisfying light restrictions. Actually, as in most topological investigations of measure preserving transformations, the main property needed is that a bounded open set can not be carried into a subset of itself such that the difference of the two sets contains interior points. I t is shown that there are compact continua of assorted sizes which contain the fixed point and which are carried into themselves by the transformation. Such continua might, for example, be solid spheres # i 2 + • • • +xn ^r as in the case of an orthogonal transformation. On the other hand they might be arcs as in the case of the transformation X\ :=z ZXi) Xn~i == Z#n_i , Xn =s 1 / £ Xni where continua of the type described are intervals on the #n-axis which include the origin. The results also show that there are certain points near the fixed point which remain near it under indefinite positive iteration of the transformation. We use the symbol Z7"" for the set T~(U)y and so on. THEOREM 1. Let T be a measure preserving homeomorphism of En onto itself', and let A be a compact connected set such that T(A)CZA. Then if U is an open set with compact closure which includes A, there exists a compact connected set K of which A is a proper subset and such that K isinV~andT(K)CK.