Richard D. Dunphy: The Measure of Honor

On September 20, 2013, I had the pleasure of attending a town hall meeting at Gettysburg College featuring three members of Congressional Medal of Honor Society (CMOHS). Each had served our country with bravery and valor, each had gone above and beyond the call of duty, and each had earned the same medal as the man whose life I have been exploring for the past several months. [excerpt

On `observable' Li-Yorke tuples for interval maps

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect to Lebesgue), then the $d$-tuples have Lebesgue full measure, but if $T$ preserves an infinite absolutely continuous measure, the situation becomes more interesting. Taking the family of Manneville-Pomeau maps as example, we show that for any $d \ge 2$, it is possible that the set of Li-Yorke $d$-tuples has full Lebesgue measure, but the set of Li-Yorke $d+1$-tuples has zero Lebesgue measure

Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk

We prove that, for every $\alpha > -1$, the pull-back measure $\phi ({\cal A}_\alpha)$ of the measure $d{\cal A}_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit disk \D, by every analytic self-map \phi \colon \D \to \D is not only an $(\alpha + 2)$-Carleson measure, but that the measure of the Carleson windows of size \eps h is controlled by \eps^{\alpha + 2} times the measure of the corresponding window of size $h$. This means that the property of being an $(\alpha + 2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces

Canonical sequences of optimal quantization for condensation measures

Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a measure $P$ is called a condensation measure, or an an inhomogeneous self-similar measure, associated with the condensation system $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \nu)$. Let $D(\mu)$ denote the quantization dimension of a measure $\mu$ if it exists. Let $\kappa$ be the unique number such that $(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}+(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}=1$. In this paper, we have considered four different self-similar measures $\nu:=\nu_1, \nu_2, \nu_3, \nu_4$ satisfying $D(\nu_1)>\kappa$, $D(\nu_2)\kappa$, and $D(\nu_4)=\kappa$. For each measure $\nu$ we show that there exist two sequences $a(n)$ and $F(n)$, which we call as canonical sequences. With the help of the canonical sequences, we obtain a closed formula to determine the optimal sets of $F(n)$-means and $F(n)$th quantization errors for the condensation measure $P$ for each $\nu$. Then, we show that for each measure $\nu$ the quantization dimension $D(P)$ of the condensation measure $P$ exists, and satisfies: $D(P)=\max\{\kappa, D(\nu)\}$. Moreover, we show that for $D(\nu_1)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; on the other hand, for $\nu=\nu_2, \nu_3, \nu_4$, the $D(P)$-dimensional lower quantization coefficient is infinity. This shows that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients can be either finite, positive and unequal, or it can be infinity

Riesz s-Equilibrium Measures on d-Dimensional Fractal Sets as s Approaches d

Let $A$ be a compact set in \Rp of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes \Is(\mu):=\iint\Rk{x}{y}{s}d\mu(y)d\mu(x) over all such probability measures. In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then $\mu^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below