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2020, Volume 40, Issue 11: 6329-6350. Doi: 10.3934/dcds.2020281

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Matching for a family of infinite measure continued fraction transformations

1.
Department of Mathematics, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands
2.
John Cabot University, 00165 Roma, Italy, Via della Lungara 233, 00165 Roma, Italy

Received: December 2019

Revised: May 2020

Published: July 2020

The third author is supported by the NWO TOP-Grant No. 614.001.509

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Abstract

As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $ \sigma $-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

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Mathematics Subject Classification: Primary: 11K50, 37A05, 37A35, 37A40, Secondary: 11A55, 28D05, 37E05.

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Figure 1. The Gauss map $ G $ and the flipped map $ R = 1-G $ in (a) and (b). The folded $ \alpha $-continued fraction map $ \hat S_\alpha $ and the flipped $ \alpha $-continued fraction map $ T_\alpha $ for $ \alpha < \frac12 $ in (c) and (e) and for $ \alpha > \frac12 $ in (d) and (f)

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Figure 4. Numerical simulations of $ \mathcal{D}_\alpha $ for $ \alpha > \frac{1}{2}\sqrt{2} $

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Figure 2. The transformation $ \mathcal{T}_{\alpha} $ maps areas on the top to areas on the bottom with the same color or pattern

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Figure 3. The maps $ \mathcal T_\alpha $ for various values of $ \alpha $. Areas on the left are mapped to areas on the right with the same color or pattern

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Figure 5. Maps $ T_\alpha $ and $ T_{\alpha'} $ that are not $ c $-isomorphic for any $ c \in \mathbb (0, \infty] $

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Table 1. Invariant densities for $ \alpha \in \big[\frac12, \frac12 \sqrt 2 \big] $

$ \boldsymbol{\alpha} $	Density $ \boldsymbol{f_\alpha} $
$ [\frac12, g) $	$ \frac{1}{1-x} \mathbf 1_{[1-\alpha, \alpha]}(x) + \frac1{x(1-x)} \mathbf 1_{[\alpha, \frac{1-\alpha}{\alpha} ]} (x)+ \frac{x^2+1}{x(1-x^2)} \mathbf 1_{[\frac{1-\alpha}{\alpha},1]} (x) $
$ [g, \frac23) $	$ \big(\frac{1}{1-x} + \frac{1}{x+\frac{1}{g-1}}\big)\mathbf 1_{[1-\alpha, \frac{2\alpha-1}{\alpha}]} (x) + \frac1{1-x} \mathbf 1_{[\frac{2\alpha-1}{\alpha}, \alpha ]} (x)+ $
	$ + \big(\frac{1}{1-x} + \frac{1}{x} -\frac{1}{x+\frac{1}{g}}\big) \mathbf 1_{[\alpha,\frac{2\alpha-1}{1-\alpha}]} (x)+ \frac{x^2+1}{x(1-x^2)} \mathbf 1_{[\frac{2\alpha-1}{1-\alpha},1]} (x) $
$ [\frac23, \frac12 \sqrt{2}] $	$ \big(\frac{1}{1-x} + \frac{1}{x+\frac{1}{g-1}}\big)\mathbf 1_{[1-\alpha, \frac{2\alpha-1}{\alpha}]} (x) + \frac1{1-x} \mathbf 1_{[\frac{2\alpha-1}{\alpha}, \alpha ]} (x) + \big(\frac{1}{1-x} + \frac{1}{x} -\frac{1}{x+\frac{1}{g}}\big) \mathbf 1_{[\alpha,\frac{1-\alpha}{2\alpha-1}]} (x)+ $
	$ +\big(\frac{1}{1-x}+\frac{1}{x+1} -\frac{1}{x+ \frac1g} + \frac{1}{x} - \frac{1}{x+\frac{1}{g+1}}\big) \mathbf 1_{[\frac{1-\alpha}{2\alpha-1},1]} (x) $

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