Observable Lyapunov irregular sets for planar piecewise expanding maps

For any integer $r$ with $1\le r<\mathrm{\infty }$, we present a one-parameter family $F_{\sigma }$ $(0<\sigma <1)$ of 2-dimensional piecewise ${\mathcal{C}}^r$ expanding maps such that each $F_{\sigma }$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).