# Power dissipation in fractal Feynman-Sierpinski AC circuits

## TL;DR

This paper investigates power dissipation in fractal AC circuits, specifically the Feynman-Sierpinski ladder, analyzing how power is dissipated and propagates at certain frequencies using mathematical limits and measures.

## Contribution

It introduces a novel analysis of power dissipation in fractal networks, constructing a measure that describes dissipation on the fractal structure.

## Key findings

- Power dissipation is characterized as a limit of quadratic forms.
- The power dissipation measure is continuous and singular relative to a Hausdorff measure.
- The study reveals wave propagation and dissipation phenomena at specific frequencies.

## Abstract

This paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non-dissipative elements. In particular, we analyze the so-called Feynman-Sierpinski ladder, a fractal AC circuit motivated by Feynman's infinite ladder, that exhibits power dissipation and wave propagation for some frequencies. Power dissipation in this circuit is obtained as a limit of quadratic forms, and the corresponding power dissipation measure associated with harmonic potentials is constructed. The latter measure is proved to be continuous and singular with respect to an appropriate Hausdorff measure defined on the fractal dust of nodes of the network.