We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.

中文翻译：

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分段展开图的熵公式和熵的连续性

我们考虑具有不变概率测度的有限维空间中的某些类别的分段展开图，这些不变测度相对于Lebesgue测度是绝对连续的。我们导出了此类度量的熵公式，并使用此熵公式为某些参数化地图族中的参数提供了该熵连续性的充分条件。我们将结果应用于一个经典的一维帐篷图族和一个二维图族，当一个同构相切由一个三维微分形族展开时，它作为返回图的极限而出现。