The entropy of α-continued fractions: numerical results

Carminati, Carlo; Marmi, Stefano; Profeti, A.; Tiozzo, Giulio

doi:10.1088/0951-7715/23/10/005

Cited by 13 publications

(17 citation statements)

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“…This is due to the fact that the natural extension has no finite rectangular structure when α ranges in a matching interval (see [27]). We conjecture that, as in the case we examined in this paper, on a matching interval densities are piecewise continuous, with discontinuity points located on the forward images of the endpoints (before matching occurs), while the branches of these densities are fractional transformation which move smoothly with the parameter (see also [9], Conj. 5.3).…”

Section: Comparison With Nakada's α-Continued Fractions and Open Quesmentioning

confidence: 75%

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy

Carminati

Isola

Tiozzo

2018

Trans. Amer. Math. Soc.

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We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)continued fractions.We provide an explicit construction of the bifurcation locus E KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of Kα is analytic outside the bifurcation set but not differentiable at points of E KU , and that the entropy is monotone as a function of the parameter.Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

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Section: Comparison With Nakada's α-Continued Fractions and Open Quesmentioning

confidence: 75%

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy

Carminati

Isola

Tiozzo

2018

Trans. Amer. Math. Soc.

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“…We will see that Θ maps F to itself, and that (ζ Θ n (v) ) n≥0 is a sequence of rapidly converging quadratic numbers; see also [CMPT10]. Therefore, we define…”

Section: Resultsmentioning

confidence: 99%

Natural extensions and entropy ofα-continued fractions

2012

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We construct a natural extension for each of Nakada's α-continued fraction transformations and show the continuity as a function of α of both the entropy and the measure of the natural extension domain with respect to the density function (1 + xy) −2 . For 0 < α ≤ 1, we show that the product of the entropy with the measure of the domain equals π 2 /6. We show that the interval (3 − √ 5)/2 ≤ α ≤ (1 + √ 5)/2 is a maximal interval upon which the entropy is constant. As a key step for all this, we give the explicit relationship between the α-expansion of α − 1 and of α.

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“…other than the one given in the previous point). Then An immediate corollary is the explicit description of the orbit of the pseudocenter which explains an empirical rule given in [1]. Corollary 3.4.…”

Section: Anatomy Of Maximal Orbitsmentioning

confidence: 94%

“…A numerical study of the conjecture has been carried out in [1]: the goal of this paper is to prove the existence of the structures numerically observed there, thus proving conjecture 1.1.…”

Section: Introductionmentioning

confidence: 92%