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Giulio Tiozzo

Publications and source records attributed to Giulio Tiozzo.

At least 19 recordsLinked to original sources
arXiv2025

Random walks on cocompact Fuchsian and Kleinian groups

The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $\Gamma$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups.

math.DS↗
arXiv2025

Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

We consider symmetric random walks on discrete, Zariski-dense subgroups $\Gamma$ of a semisimple Lie group $G$ with Property (T). We prove that if $\Gamma$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank.

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arXiv2024

A cell decomposition for marked cycle curves

We describe a family $\textrm{Cyc}_p(\mathcal{F})$ of marked cycle curves that parameterize the cycles of period $p$ of a given family $\mathcal{F}$ of dynamical systems. We produce algorithms to compute a canonical cell decomposition for the marked cycle curves over the family $\textrm{Per}_1(0)$ of quadratic polynomials as well as over the family $\textrm{Per}_2(0)$ of quadratic rational maps with a critical 2-cycle. We obtain formulas for the number of $d$-cells in these decompositions, giving rise to e.g. a formula for their genus.

math.DS↗
arXiv2024

Roots of Alexander polynomials of random positive 3-braids

Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.

math.GT↗
arXiv2023

Genericity of contracting geodesics in groups

Let G be a finitely generated group and Cay(G, S) be the Cayley graph of G with respect to a finite generating set S. We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in Cay(G, S).

math.GT↗
arXiv2022

The Poisson boundary of hyperbolic groups without moment conditions

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.

math.GR↗
arXiv2022

Sublinearly Morse Boundary I: CAT(0) Spaces

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that it is QI-invariant and metrizable. This is to say, the $κ$-Morse boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it is shown in the Appendix that the Poisson boundary of random walks is naturally identified with the $\sqrt{t \log t}$--boundary.

math.GT↗
arXiv2021

Master Teapots and Entropy Algorithms for the Mandelbrot Set

We construct an analogue of W. Thurston's "Master teapot" for each principal vein in the Mandelbrot set, and generalize geometric properties known for the corresponding object for real maps. In particular, we show that eigenvalues outside the unit circle move continuously, while we show "persistence" for roots inside the unit circle. As an application, this shows that the outside part of the corresponding "Thurston set" is path connected. In order to do this, we define a version of kneading theory for principal veins, and we prove the equivalence of several algorithms that compute the core entropy.

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arXiv2021

Metrics on trees I. The tower algorithm for interval maps

We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the other hand, the theory of piecewise linear models coming from kneading theory. We prove that the algorithm converges for unimodal maps of high entropy, and provide examples where it does not converge for unimodal maps of low entropy and multimodal maps of higher degree.

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arXiv2021

A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

For geometrically finite group actions on hyperbolic metric spaces and under certain assumptions on the growth of parabolic subgroups, we prove a global shadow lemma for Patterson-Sullivan measures, as well as a Dirichlet-type theorem and a logarithm law for excursion of geodesics into cusps. We then apply these results to geometrically finite quotients of strictly convex Hilbert geometries with C^1 boundary.

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arXiv2021

The fundamental inequality for cocompact Fuchsian groups

We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff dimension of the hitting measure is strictly less than 1. A similar statement is proven for Coxeter groups. Along the way, we prove for cocompact Fuchsian groups a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.

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arXiv2021

The bifurcation locus for numbers of bounded type

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 behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. 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 a corresponding section of the bifurcation set E.

math.DS↗
arXiv2021

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2$ are mutually singular.

math.DS↗
arXiv2021

Random walks, WPD actions, and the Cremona group

We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show that the random walk satisfies a form of asymptotic acylindricality, and we use this to show that the normal closure of random elements yields almost surely infinitely many different normal subgroups. Moreover, the probability that the normal closure is free tends to 1 if and only if the maximal normal subgroup coincides with the center of the group. We apply such techniques to the Cremona group, thus obtaining that the dynamical degree of random Cremona transformations grows exponentially fast, producing many different normal subgroups, and identifying the Poisson boundary. We also give a new identification of the Poisson boundary of Out(F_n). Our methods give bounds on the rates of convergence for these results.

math.GT↗
arXiv2020

Sublinearly Morse Boundary II: Proper geodesic spaces

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space $X$ and any sublinear function $\kappa$, we construct a boundary for $X$, denoted $\mathcal{\partial}_{\kappa} X$, that is quasi-isometrically invariant and metrizable. As an application, we show that when $G$ is the mapping class group of a finite type surface, or a relatively hyperbolic group, then with minimal assumptions the Poisson boundary of $G$ can be realized on the $\kappa$-Morse boundary of $G$ equipped the word metric associated to any finite generating set.

math.GT↗