[Paper Review] Hausdorff dimension, Mean quadratic variation of infinite self-similar measures

Zu‐Guo Yu, Fu-Yao Ren+1 more|ArXiv.org|Dec 24, 1998

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper derives the Hausdorff dimension of infinite self-similar sets under weaker conditions than previous work by Riedi and Mandelbrot, and establishes the asymptotic behavior of the mean quadratic variation of infinite self-similar measures under additional hypotheses. It extends classical results in fractal geometry to infinite iterated function systems with improved generality.

ABSTRACT

Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K which is the invariant compact set of infinite contractive similarities {S_j(x)} satisfying open set condition is obtained. It is proved (under some additional hypotheses) that the mean quadratic variation of infinite self-similar measure is of asymptotic property.

Motivation & Objective

• To generalize the computation of Hausdorff dimension for infinite self-similar sets beyond the constraints of Riedi and Mandelbrot's original work.
• To investigate the asymptotic properties of the mean quadratic variation of infinite self-similar measures.
• To establish the dimension formula under the open set condition with weaker assumptions than previously required.
• To provide a theoretical foundation for the statistical behavior of infinite self-similar measures through mean quadratic variation.

Proposed method

• The authors use the open set condition to ensure non-overlapping structure in the infinite self-similar set construction.
• They apply techniques from classical analysis and measure theory to derive the Hausdorff dimension of the invariant compact set generated by infinite contractive similarities.
• The mean quadratic variation is analyzed asymptotically using probabilistic and analytic tools under additional hypotheses.
• The proof relies on recursive structure and scaling properties inherent in self-similar measures.
• The authors leverage known results on dimension theory and extend them to infinite iterated function systems.

Experimental results

Research questions

• RQ1 What is the Hausdorff dimension of an infinite self-similar set when the similarities satisfy the open set condition under weaker assumptions than Riedi and Mandelbrot?
• RQ2 How does the mean quadratic variation of an infinite self-similar measure behave asymptotically?
• RQ3 Under what conditions can the asymptotic mean quadratic variation be rigorously established for such measures?
• RQ4 Can the dimension formula be extended to infinite systems without requiring strong separation or contraction conditions?

Key findings

• The Hausdorff dimension of the infinite self-similar set K is computed under weaker conditions than those assumed by Riedi and Mandelbrot.
• The mean quadratic variation of the infinite self-similar measure exhibits a well-defined asymptotic property under additional hypotheses.
• The result confirms that the dimension formula remains valid even when the standard assumptions on contraction ratios and separation are relaxed.
• The paper provides a theoretical basis for understanding the statistical and geometric behavior of infinite self-similar measures through their quadratic variation.