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Giulio Tiozzo

Yale University, Mathematics, Faculty Member

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Pontificia Universidad Catolica de Chile

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Papers by Giulio Tiozzo

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

Dynamics of continued fractions and kneading sequences of unimodal maps

by Carlo Carminati, S. Isola, and Giulio Tiozzo

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.

CONTINUED FRACTIONS WITH SL(2, Z)-BRANCHES: COMBINATORICS AND ENTROPY

by Carlo Carminati and Giulio Tiozzo

We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branc... more 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.

$Research paper thumbnail of Tuning and plateaux for the entropy of $\alpha$-continued fractions$

Tuning and plateaux for the entropy of $\alpha$-continued fractions

by Carlo Carminati and Giulio Tiozzo

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

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

A canonical thickening of Q and the dynamics of continued fractions

by Carlo Carminati and Giulio Tiozzo

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.

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

The entropy of alpha-continued fractions: numerical results

by Carlo Carminati and Giulio Tiozzo

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.

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

The entropy of alpha-continued fractions: numerical results

by Carlo Carminati and Giulio Tiozzo

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.

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

The entropy of alpha-continued fractions: numerical results

by Carlo Carminati, A. Profeti, and Giulio Tiozzo

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.

Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Given a measure on the Thurston boundary and a basepoint in Teichmüller space, one can pick a geo... more Given a measure on the Thurston boundary and a basepoint in Teichmüller space, one can pick a geodesic ray joining the basepoint to a randomly chosen point on the boundary. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a finitely supported distribution. For any geodesic ray, we compute the ratio between the word metric and the relative metric of approximating mapping class group elements, and prove that this ratio tends to infinity along Lebesgue-typical geodesic, while the limit is finite along geodesics which are typical with respect to harmonic measure. As a corollary, we establish singularity of harmonic measure. The same proof works for Fuchsian groups with a cusp.

Generalised continuation by means of right limits

Several theories have been proposed to generalise the concept of analytic continuation to holomor... more Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon's work on right limits of power series, Baladi-Marmi-Sauzin recently introduced the notion of renascent right limit and rrl-continuation.

$Research paper thumbnail of Tuning and plateaux for the entropy of $α$-continued fractions$

Tuning and plateaux for the entropy of $α$-continued fractions

The entropy h(Tα) of α-continued fraction transformations is known to be locally monotone outside... more The entropy h(Tα) of α-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set E. We will exploit the explicit description of the fractal structure of E to investigate the self-similarities displayed by the graph of the function α → h(Tα). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

Sublinear deviation between geodesics and sample paths

We give a proof of the sublinear tracking property for sample paths of random walks on various gr... more We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmüller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely generated groups.

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

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 ...

The bifurcation locus for numbers of bounded type

arXiv preprint arXiv:1109.0516, Jan 1, 2011

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of num... more We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set E.

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

The entropy of α-continued fractions: numerical results

Nonlinearity, Jan 1, 2010

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

The entropy of alpha-continued fractions: analytical results

Arxiv preprint arXiv:0912.2379, Jan 1, 2009

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

A canonical thickening of Q and the dynamics of continued fractions

Arxiv preprint arXiv:1004.3790, Jan 1, 2010

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

Dynamics of continued fractions and kneading sequences of unimodal maps

Arxiv preprint arXiv: …, Jan 1, 2010

Dinamica delle frazioni continue e teorema del limite centrale

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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