We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.

中文翻译：

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具有平坦点和不同临界指数的圆图的相变

我们研究具有平坦间隔的圆图，其中平坦点两个边界点的临界指数可能不同。这种系统的空间被划分为两个相连的部分，它们的公共边界仅取决于临界指数。在该边界处，系统的几何形状发生相变。与之前的方法不同，这是通过研究重整化算子的渐近行为来实现的。