Fractal Symmetry & Noether's Theorem: Chaos & Conservation

[Michele Zappano]

I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

 

[Dr. Courtney]

I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

One needs to distinguish between physical symmetries and mathematical symmetries.

Noether's Thm can be stated informally as:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

I haven't read the Strogatz book, but two questions that come to mind are:

Is the purported symmetry in fractals a continuous symmetry?

Is the purported symmetry in fractals a physical symmetry (really a perfect symmetry in a physical realization)?

A lot of purported symmetries in physical systems are neither exact nor continuous.

 

[Andy Resnick]

I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

This is at the edge of my understanding: scale invariance (self similarity) is associated with conformal symmetry:

https://en.wikipedia.org/wiki/Conformal_symmetry

These are typically associated with phase transitions, but I think you can also generate a conserved energy-momentum tensor from this.

 

[Michele Zappano]

This is at the edge of my understanding: scale invariance (self similarity) is associated with conformal symmetry:

https://en.wikipedia.org/wiki/Conformal_symmetry

These are typically associated with phase transitions, but I think you can also generate a conserved energy-momentum tensor from this.

perfect, thanks