Arithmetic mod 1

Question

I was reading the news and a topological data paper was out about arithmetic mod 1, it had persistence graphs of some functions. Isn't arithmetic mod 1 only with decimals? Why is it so popular in ergodic theory?

Answer 1

Isn't arithmetic mod 1 only with decimals?

This arithmetic has nothing to do with the representation. One builds equivalence classes $$ x\equiv y \qquad\Leftrightarrow\qquad x - y = k\cdot1\ \text{ for some } k\in\Bbb Z $$ i.e. one only considers sets of numbers, and in each set the numbers differ by an integral amount. You can use the interval $[0,1)\subset\Bbb R$ to uniquely represent these classes, and you can define well-defined operations $+$, $-$, $\times$ as obvious. Computation is "perform operation and throw away the integral part". For example: $$ 1\equiv 2$$

$$ \tfrac13 -\tfrac12\equiv\tfrac56$$

$$ 4\cdot0.5 \equiv 2\cdot0.5 = 0.5 + 0.5\equiv 0$$

The last means there are zero divisors, and you cannot define division for rational numbers mod 1 in a sensible way.

Bit more intuitive is to represent that ring as rotations of a circle by some angle $\phi$:

$$ \phi\equiv \psi \qquad\Leftrightarrow\qquad \phi - \psi = k\cdot2\pi\ \text{ for some } k\in\Bbb Z $$

The dynamics is already non-trivial, it's the Julia-set for $z\mapsto z^2$, see for example Dynamik von z→z².

Answer 2

I think $mod \: 1$ is simply defined as follw: $0$ if $n$ is an intger else {n} (where $\{\}$ is the decimal part) for $n$ mot an integer.