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Pietro Poggi-Corradini

Publications and source records attributed to Pietro Poggi-Corradini.

42 records · Page 3Linked to original sources
arXiv2006

On a theorem of Landau and Toeplitz

The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carathéodory, 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 streamlined proof of this result and also extend it to include bounds on the growth of the maximum modulus.

math.CV↗
arXiv2006

Mapping properties of analytic functions on the disk

There is a universal constant $0 0$ so that the Euclidean area, counting multiplicity, of the portion of $f(\D)$ which lies over the disk $D(f(0),M)$, centered at $f(0)$ and of radius $M$, is strictly less than the area of $D(f(0),M)$. Then $f$ must send $r_0\bar{\D}$ into $D(f(0),M)$. This answers a conjecture of Don Marshall.

math.CV↗
arXiv2004

Pointwise convergence on the boundary in the Denjoy-Wolff Theorem

If $ϕ$ is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy-Wolff Theorem predicts the existence of a point $p$ with $|p|\leq 1$ such that the iterates $ϕ_{n}$ 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 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 $ϕ$. We show that when $|p|<1$(elliptic case), pointwise a.e. convergence holds if and only if $ϕ$ is not an inner function. When $|p|=1$ things are more delicate. We show that when $ϕ$ is hyperbolic or type I parabolic, then pointwise a.e. convergence holds always. The last case, type II parabolic remains open at this moment, but we conjecture the answer to be as in the elliptic case.

math.CV↗
arXiv2003

On Valiron's Theorem

This is a survey on Valiron's Theorem about the convergence properties of orbits of analytic self-maps of the disk of hyperbolic type and related questions in one and several variables.

math.CV↗
arXiv2003

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. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, 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.

math.CV↗