Giulio Tiozzo

Sublinear deviation between geodesics and sample paths

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 Teichmueller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely generated groups.

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 a closed, totally disconnected set . We will exploit the explicit description of the fractal structure of 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.

The entropy of α-continued fractions: numerical results

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as α-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.

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

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 fractions, thus proving a conjecture of Nakada and Natsui.

Counting loxodromics for hyperbolic actions

Let

Word length statistics for Teichmuller geodesics and singularity of harmonic measure

G \curvearrowright X

Generalised continuation by means of right limits

be a nonelementary action by isometries of a hyperbolic group

The local Hölder exponent for the entropy of real unimodal maps

G

Random walks on weakly hyperbolic groups

on a hyperbolic metric space

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

X