Given a Sobolev homeomorphism f ? W 2,1 in the plane we find piecewise quadratic that approximates it up to set of ? measure. We show this map can be approximated by diffeomorphisms norm on set.

We investigate how planar Sobolev-Orlicz homeomorphisms map sets of Hausdorff dimension less than two. With the correct gauge functions the generalized Hausdorff measures of the image sets are shown to be zero.

It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discontinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of maximal entropy.

We give a direct proof of a result of Earle, Gardiner and Lakic, that is, Kobayashi’s metric and Teichmüller’s metric coincide with each other on the Teichmüller space of symmetric circle homeomorphisms.

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space W 1,2 and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.

We present a classification of m-strict limits (i.e. and |D1fk|(Ω)+|D2fk|(Ω)→|D1f|(Ω)+|D2f|(Ω)) planar BV homeomorphisms; class previously studied by the authors S. Hencl in [6]. There it was shown that such mappings allow for cavitations fractures singularities but fulfil suitable generalization INV condition. As pointed out J. Ball [3], these features are physically expected limit configurati...