[Paper Review] On a fixed point in the metric space of normalized Hausdorff moment sequences

Christian Berg, Maryam Beygmohammadi|arXiv (Cornell University)|Mar 19, 2010

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes the existence and uniqueness of a fixed point in the metric space of normalized Hausdorff moment sequences under the transformation $ T((x_n)) = \left(\frac{1}{1 + \sum_{i=1}^n x_i}\right) $. Using Banach's fixed point theorem on a compact convex subset $ \mathcal{C} \subset \mathcal{K} $, it proves that the transformation is a contraction with Lipschitz constant $ \frac{8}{9} $, ensuring global attractivity of the fixed point, which is shown to be a normalized Hausdorff moment sequence with a convex, increasing density.

ABSTRACT

We show that the transformation (x_n)_{n\ge 1} o (1/(1+x_1+...+x_n))_{n\ge 1} of the compact set of sequences (x_n)_{n\ge 1} of numbers from the unit interval [0,1] has a unique fixed point, which is attractive. The fixed point turns out to be a Hausdorff moment sequence studied in papers by Berg and Durán in 2008.

Motivation & Objective

• To establish the existence and uniqueness of a fixed point for the transformation $ T((x_n)) = \left(\frac{1}{1 + \sum_{i=1}^n x_i}\right) $ on the space of sequences in $[0,1]^\mathbb{N}$.
• To prove that this fixed point is globally attractive in the metric space $ \mathcal{K} = [0,1]^\mathbb{N} $, extending prior results restricted to the subset of Hausdorff moment sequences.
• To demonstrate that the fixed point is a normalized Hausdorff moment sequence, confirming its moment representation and structural properties.
• To show that the transformation $ T $ is a contraction on a compact convex subset $ \mathcal{C} \subset \mathcal{K} $, enabling application of Banach's fixed point theorem.

Proposed method

• Define the transformation $ T: \mathcal{K} \to \mathcal{K} $ by $ T((x_n))_n = \frac{1}{1 + \sum_{i=1}^n x_i} $, acting on the compact product space $ \mathcal{K} = [0,1]^\mathbb{N} $.
• Equip $ \mathcal{K} $ with the product topology induced by the metric $ d((a_n),(b_n)) = \sum_{n=1}^\infty 2^{-n} |a_n - b_n| $, making it a Fréchet space.
• Introduce the compact convex subset $ \mathcal{C} = \{(a_n) \in \mathcal{K} \mid a_1 \geq \frac{1}{2}\} $, which is invariant under $ T $.
• Prove that $ T $ is a contraction on $ \mathcal{C} $ with Lipschitz constant $ \frac{8}{9} $, using the inequality $ d(T(a_n), T(b_n)) \leq \frac{8}{9} d((a_n), (b_n)) $ for all $ (a_n), (b_n) \in \mathcal{C} $.
• Apply Banach’s fixed point theorem to $ T|_{\mathcal{C}} $, guaranteeing a unique fixed point in $ \mathcal{C} $, which must be the unique fixed point in $ \mathcal{K} $.
• Verify that the fixed point satisfies the recurrence $ m_{n+1}^2 + \frac{m_{n+1}}{m_n} - 1 = 0 $, and confirm it is a normalized Hausdorff moment sequence via known characterization theorems.

Experimental results

Research questions

• RQ1 Does the transformation $ T((x_n)) = \left(\frac{1}{1 + \sum_{i=1}^n x_i}\right) $ have a unique fixed point in the space $ \mathcal{K} = [0,1]^\mathbb{N} $?
• RQ2 Is this fixed point globally attractive under iteration of $ T $, regardless of the starting point in $ \mathcal{K} $?
• RQ3 Can the fixed point be characterized as a normalized Hausdorff moment sequence, and what are the properties of its associated measure?
• RQ4 What is the best Lipschitz constant for $ T $, and does $ T $ act as a contraction on any invariant subset of $ \mathcal{K} $?

Key findings

• The transformation $ T $ has a unique fixed point $ (m_n) $ in $ \mathcal{K} $, satisfying $ (1 + m_1 + \cdots + m_n)m_n = 1 $ for all $ n \geq 1 $.
• The fixed point is globally attractive: for any initial sequence in $ \mathcal{K} $, the iterates $ T^n $ converge to $ (m_n) $ in the metric topology.
• The fixed point $ (m_n) $ is a normalized Hausdorff moment sequence, i.e., $ m_n = \int_0^1 x^n \, d\tau(x) $ for a probability measure $ \tau $ on $[0,1]$ with increasing and convex density.
• The transformation $ T $ is a contraction on the subset $ \mathcal{C} = \{(a_n) \in \mathcal{K} \mid a_1 \geq \frac{1}{2}\} $ with Lipschitz constant $ \frac{8}{9} $, enabling application of Banach’s fixed point theorem.
• The diameter of the set $ \mathcal{H} $ of normalized Hausdorff moment sequences is 1, and the only pair achieving this diameter is $ \{\underline{0}, \underline{1}\} $.
• The fixed point sequence is explicitly computed as $ m_1 = \frac{-1 + \sqrt{5}}{2} \approx 0.618 $, and higher terms follow from the recurrence $ m_{n+1}^2 + \frac{m_{n+1}}{m_n} - 1 = 0 $.