Elementary Proof of Existence of a Subordinate Smooth Partition of Unity on Smooth Manifolds

We give in particular an elementary proof of the existence of a smooth partition of unity subordinate to any given open cover for smooth manifolds. As a side note, given are also two elementary proofs of the existence of a subordinate partition of unity for topological manifolds. These in particular fill a gap in the related literature.

Invariant measures for interval translations and some other piecewise continuous maps

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.

On Primitive 3-smooth Partitions of $n$

A primitive 3-smooth partition of $n$ is a representation of $n$ as the sum of numbers of the form $2^a 3^b$, where no summand divides another. Partial results are obtained in the problem of determining the maximal and average order of the number of such representations. Results are also obtained regarding the size of the terms in such a representation, resolving questions of Erdős and Selfridge.

Smooth Markov partitions and toral automorphisms

We show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear. Orbits of affine disks in the boundary are used to construct a basis of 2-dimensional invariant toral subgroups, and then the product decomposition of a covering follows easily.