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 o...

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Bibliographic Details
Published in:Dynamical systems (London, England) England), 2024-10, Vol.39 (4), p.725-758
Main Authors: Cockram, Noah, Ashwin, Peter, Rodrigues, Ana
Format: Article
Language:English
Subjects:
continued fractions
Half planes
Parameters
Piecewise isometry
renormalization
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Summary: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 tied to Diophantine properties of the parameters, and subsequently prove the infinite renormalizability of the first return map for these parameters.
ISSN:1468-9367
1468-9375
DOI:10.1080/14689367.2024.2378297