Giulio Tiozzo | Yale University (original) (raw)

Papers by Giulio Tiozzo

$Research paper thumbnail of Dynamics of continued fractions and kneading sequences of unimodal maps$

In this paper we construct a correspondence between the parameter spaces of two families of one-d... more In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

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$Research paper thumbnail of Tuning and plateaux for the entropy of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>-continued fractions$

The entropy h(Talpha)h(T_\alpha)h(Talpha) of alpha\alphaalpha-continued fraction transformations is known to be locally m... more The entropy h(Talpha)h(T_\alpha)h(Talpha) of alpha\alphaalpha-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set EE\EEEE. We will exploit the explicit description of the fractal structure of EE\EEEE to investigate the self-similarities displayed by the graph of the function alphamapstoh(Talpha)\alpha \mapsto h(T_\alpha)alphamapstoh(Talpha). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

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$Research paper thumbnail of A canonical thickening of Q and the dynamics of continued fractions$

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadrati... more We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of alpha-continued fractions, thus proving a conjecture of Nakada and Natsui.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

Nonlinearity, 2010

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

Nonlinearity, 2010

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of Tuning and plateaux for the entropy of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>-continued fractions$

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$Research paper thumbnail of A canonical thickening of Q and the entropy of α-continued fraction transformations$

Ergodic Theory and Dynamical Systems, Jan 1, 2011

Abstract. We construct a countable family of open intervals contained in (0, 1] whose endpoints a... more Abstract. We construct a countable family of open intervals contained in (0, 1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued ...

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arXiv preprint arXiv:1109.0516, Jan 1, 2011

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$Research paper thumbnail of The entropy of α-continued fractions: numerical results$

Nonlinearity, Jan 1, 2010

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$Research paper thumbnail of The entropy of alpha-continued fractions: analytical results$

Arxiv preprint arXiv:0912.2379, Jan 1, 2009

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$Research paper thumbnail of A canonical thickening of Q and the dynamics of continued fractions$

Arxiv preprint arXiv:1004.3790, Jan 1, 2010

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$Research paper thumbnail of Dynamics of continued fractions and kneading sequences of unimodal maps$

Arxiv preprint arXiv: …, Jan 1, 2010

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Università di Pisa. Home Unipi. banca dati delle tesi e dissertazioni accademiche elettroniche. a... more Università di Pisa. Home Unipi. banca dati delle tesi e dissertazioni accademiche elettroniche. a cura del Sistema bibliotecario di ateneo. Tesi etd-03102008-135356. Tipo di tesi, Tesi di laurea specialistica. Autore, TIOZZO, GIULIO. URN, etd-03102008-135356. ...

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$Research paper thumbnail of Dynamics of continued fractions and kneading sequences of unimodal maps$

In this paper we construct a correspondence between the parameter spaces of two families of one-d... more In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

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$Research paper thumbnail of Tuning and plateaux for the entropy of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>-continued fractions$

The entropy h(Talpha)h(T_\alpha)h(Talpha) of alpha\alphaalpha-continued fraction transformations is known to be locally m... more The entropy h(Talpha)h(T_\alpha)h(Talpha) of alpha\alphaalpha-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set EE\EEEE. We will exploit the explicit description of the fractal structure of EE\EEEE to investigate the self-similarities displayed by the graph of the function alphamapstoh(Talpha)\alpha \mapsto h(T_\alpha)alphamapstoh(Talpha). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

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$Research paper thumbnail of A canonical thickening of Q and the dynamics of continued fractions$

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadrati... more We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of alpha-continued fractions, thus proving a conjecture of Nakada and Natsui.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

Nonlinearity, 2010

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

Nonlinearity, 2010

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of The entropy of alpha-continued fractions: numerical results$

We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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$Research paper thumbnail of Tuning and plateaux for the entropy of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>-continued fractions$

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$Research paper thumbnail of A canonical thickening of Q and the entropy of α-continued fraction transformations$

Ergodic Theory and Dynamical Systems, Jan 1, 2011

Abstract. We construct a countable family of open intervals contained in (0, 1] whose endpoints a... more Abstract. We construct a countable family of open intervals contained in (0, 1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued ...

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arXiv preprint arXiv:1109.0516, Jan 1, 2011

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$Research paper thumbnail of The entropy of α-continued fractions: numerical results$

Nonlinearity, Jan 1, 2010

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$Research paper thumbnail of The entropy of alpha-continued fractions: analytical results$

Arxiv preprint arXiv:0912.2379, Jan 1, 2009

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$Research paper thumbnail of A canonical thickening of Q and the dynamics of continued fractions$

Arxiv preprint arXiv:1004.3790, Jan 1, 2010

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$Research paper thumbnail of Dynamics of continued fractions and kneading sequences of unimodal maps$

Arxiv preprint arXiv: …, Jan 1, 2010

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Università di Pisa. Home Unipi. banca dati delle tesi e dissertazioni accademiche elettroniche. a... more Università di Pisa. Home Unipi. banca dati delle tesi e dissertazioni accademiche elettroniche. a cura del Sistema bibliotecario di ateneo. Tesi etd-03102008-135356. Tipo di tesi, Tesi di laurea specialistica. Autore, TIOZZO, GIULIO. URN, etd-03102008-135356. ...

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