A canonical thickening of ℚ and the entropy of<i>α</i>-continued fraction transformations

Carminati, Carlo; Tiozzo, Giulio

doi:10.1017/s0143385711000447

Cited by 26 publications

(43 citation statements)

References 11 publications

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“…Indeed, on a combinatorial level the structure of the real slice of the Mandelbrot set is isomorphic to the structure of the bifurcation set E for continued fractions [5], so we can use the combinatorial tools we developed in that case [10,11] to analyze the quadratic family.…”

Section: Pseudocenters and Real Hyperbolic Windowsmentioning

confidence: 99%

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Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Tiozzo

2015

Advances in Mathematics

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A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy. Indeed, we prove an "entropy formula" relating the entropy of a polynomial restricted to its Hubbard tree to the Hausdorff dimension of the set of rays landing on the corresponding vein in the Mandelbrot set. The results contribute to the recent program of W. Thurston of understanding the geometry of the Mandelbrot set via the core entropy.

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Section: Pseudocenters and Real Hyperbolic Windowsmentioning

confidence: 99%

“…For instance, in [10], a key concept is the pseudocenter of an interval, namely the (unique!) rational number with the smallest denominator.…”

Section: Pseudocenters and Real Hyperbolic Windowsmentioning

confidence: 99%

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Tiozzo

2015

Advances in Mathematics

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“…Also, if matching implies monotonicity is not clear. Certain conditions used to prove it in the case of Nakada α-expansions do not hold for our family and therefore we cannot mimic the proof of [18]. Let us first give the definition of matching.…”

Section: Entropymentioning

confidence: 99%

Invariant measures for continued fraction algorithms with finitely many digits

Kraaikamp

Langeveld

2017

Journal of Mathematical Analysis and Applications

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In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss-Kuzmin-Lévy based approximation method is used. Convergence of this method follows from [1] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N -expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N = 2 and N = 36. Interesting behavior can be observed from numerical results.

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“…The structure of the matching set for α-continued fractions is quite well understood ( [10], [6]), but in this case matching intervals with different monotonic behaviours are mixed up in a complicated way ( [12]), so even the fact that the entropy h N attains its maximum value at 1/2 is still conjectural.…”

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

confidence: 99%

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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A canonical thickening of ℚ and the entropy ofα-continued fraction transformations

Cited by 26 publications

References 11 publications

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Invariant measures for continued fraction algorithms with finitely many digits

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

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