Pietro Poggi-Corradini

Area, capacity and diameter versions of Schwarz’s Lemma

The now canonical proof of Schwarz’s Lemma appeared in a 1907 paper of Caratheodory, who attributed it to Erhard Schmidt. Since then, Schwarz’s Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc.

Shape space of achiral simplexes

We examine the general dimensional relation between the shape space of then-simplexes and the subspace of their achiral sets. It is found that the only simplexes that can be partitioned into “left-handed” and “right-handed” classes are the triangles in Euclidean 2-space.

BACKWARD-ITERATION SEQUENCES WITH BOUNDED HYPERBOLIC STEPS FOR ANALYTIC SELF-MAPS OF THE DISK

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880s to the 1980s, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

Optimal information dissemination strategy to promote preventive behaviors in multilayer epidemic networks.

Launching a prevention campaign to contain the spread of infection requires substantial financial investments; therefore, a trade-off exists between suppressing the epidemic and containing costs. Information exchange among individuals can occur as physical contacts (e.g., word of mouth, gatherings), which provide inherent possibilities of disease transmission, and non-physical contacts (e.g., email, social networks), through which information can be transmitted but the infection cannot be transmitted. Contact network (CN) incorporates physical contacts, and the information dissemination network (IDN) represents non-physical contacts, thereby generating a multilayer network structure. Inherent differences between these two layers cause alerting through CN to be more effective but more expensive than IDN. The constraint for an epidemic to die out derived from a nonlinear Perron-Frobenius problem that was transformed into a semi-definite matrix inequality and served as a constraint for a convex optimization problem. This method guarantees a dying-out epidemic by choosing the best nodes for adopting preventive behaviors with minimum monetary resources. Various numerical simulations with network models and a real-world social network validate our method.

Valiron's construction in higher dimension

We consider holomorphic self-maps ϕ of the unit ball B N in C N (N =1 , 2, 3 ,... ). In the one-dimensional case, when ϕ has no fixed points in D := B 1 and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map ϕ, and therefore, in this case, the dynamical properties of ϕ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on ϕ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation σ, which maps the ball into the right half-plane of C ,a nd solves the functional equation σ ◦ ϕ = λσ ,w here λ> 1i s the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of ϕ.

Pointwise Convergence on the Boundary in the Denjoy-Wolff Theorem

If is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy- Wol Theorem predicts the existence of a point p with |p| 1 such that the iterates n converge to p uniformly on compact subsets of the disk. Since these iterates are bounded analytic functions, there is a subset of the unit circle of full linear measure where they are all well-defined. We address the question of whether convergence to p still holds almost everywhere on the unit circle. The answer depends on the location of p and the dynamical properties of . We show that when |p| < 1(elliptic case), pointwise a.e. convergence holds if and only if is not an inner function. When |p| = 1 things are more delicate. We show that when is hyperbolic or non-zero-step parabolic, then pointwise a.e. convergence holds always. The last case, zero-step parabolic, remains open.

Hardy spaces and twisted sectors for geometric models

We study the one-to-one analytic maps a that send the unit disc into a region G with the property that AG C G for some complex number A, 0 < JAI < 1. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region G that characterize their rate of growth, i.e. we prove that a EGnP< HP if and only if G does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

AbstractThe notion of p-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for p-modulus. We show that minimal subfamilies have at most |E| elements and that these elements carry a weight related to their “importance” in relation to the corresponding p-modulus problem. When

On the Uniqueness of Classical Semiconjugations for Self-Maps of the Disk

p=2