Continuity of the entropy for monotonic mod one transformations (Q1276280)

The author considers the problem of continuity of the topological entropy \(h_{\text{top}}\) on the space of piecewise monotone transformations of \(I=[0,1]\). Two piecewise monotone transformations of \(I\) are near if the closures of their graphs in \(I^2\) are near with respect to the Hausdorff distance. It is proven that for a continuous strictly increasing function \(f:I\to\mathbb R\) the topological entropy \(h_{\text{top}}\) is continuous at the piecewise monotone transformation \(T:=f\;(\text{mod} 1)\) (such transformations \(T\) are called monotone mod one transformations), provided \(h_{\text{top}}(T)>0\). The author presents two examples of monotone mod one transformations of \(I\) at which the entropy is discontinuous. Also, he gives an example of a monotonic mod one transformation \(T\) with positive (and thus continuous at \(T\)) entropy such that the maximal measure and the pressure \(p(T,g)\) are discontinuous at \(T\) for some continuous function \(g:I\to\mathbb R\).