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Renormalization of translated cone exchange transformations

In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a geometric construction for the first return map to a neighbourhood of the vertex of the middle cone for a large class of parameters, then we show a recurrence in the first return map...