## Abstract

In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter α ∈ [g, G], where g = ^{1}_{2} (^{√}5 − 1) and G = g + 1 = 1/g are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each α, α^{∗} ∈ [g, G] the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for α ∈ [g, G], a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of α smaller than g, and that for values of α ∈ [ ^{1}_{6} (^{√}13 − 1), g] all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of α ∈ [0, G]. It is shown that if there exists an ergodic, absolutely continuous Tα-invariant measure, in any neighborhood of 0 we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.

Original language | English |
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Pages (from-to) | 2852-2888 |

Number of pages | 37 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 43 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2023 |

### Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

## Keywords

- matching
- natural extensions
- Odd continued fractions