Every Possible Cycle Length
Pyth, 11 8 bytes
.&lt;W-0zz1

A lot more boring than my previous answer.
Every number that contains a 0 digit maps to itself. Any other number maps to the number that has its digits rotated by 1. So for example:
1 -&gt; 1
10 -&gt; 10
15 -&gt; 51 -&gt; 15
104 -&gt; 104
123 -&gt; 231 -&gt; 312 -&gt; 123