Quilting natural extensions for alpha-Rosen Fractions
Cor Kraaikamp, Thomas A. Schmidt, Ionica Smeets
TL;DR
This paper constructs natural extensions for alpha-Rosen continued fractions at small alpha values by modifying the region of the standard Rosen fractions, demonstrating that the associated maps share the same entropy.
Contribution
It introduces a method to obtain natural extensions for alpha-Rosen fractions through geometric modifications, revealing entropy equivalence.
Findings
Natural extensions are constructed for small alpha values.
Modified regions lead to equal entropy among the maps.
The approach generalizes previous Rosen fraction analyses.
Abstract
We give natural extensions for the alpha-Rosen continued fractions of Dajani et al. for a set of small alpha values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions. It follows that the underlying maps have equal entropy.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Algebra and Geometry · Fractal and DNA sequence analysis