Self-similar carpets over finite fields

TL;DR

This paper explores linear dynamical systems over finite fields, demonstrating their self-similar structures and connections to tilings and number theory, with prime fields represented as self-similar carpets.

Contribution

It establishes the self-similar nature of certain finite field dynamical systems and links them to tilings and number theory, providing a new geometric perspective.

Findings

Proves self-similarity in finite field dynamical systems

Connects self-similar carpets to aperiodic tilings and Delanoy numbers

Shows prime fields can be represented as self-similar carpets

Abstract

Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a canonical presentation as sets of self-similar carpets. The multiplicative inverse corresponds to mirroring.