July 25-30, 2010

Lecture Course III:

The theory of *p*-adic numbers, developed at the end of the 19th Century,
is fundamental in modern number theory.
It is based on an alternative definition of absolute value, which derives
from a hierarchical (as opposed to sequential) arrangement of the integers.
This device opens the study of integer congruences to the methods of analysis,
and results in a number system with rather unusual properties. Recently,
*p*-adic numbers have shown up in other areas of mathematics; this course
is an introduction to the lively research area of *p*-adic dynamical systems.

SYLLABUS:

- Gauss, chaos, and digits of rationals.
*p*-adic numbers.- Linear
*p*-adic dynamics; recurrence. - Hensel's lifting; linearisation and
*p*-adic Siegel discs. - Application: planar rotations with round-off errors.

Lecture Notes |

BACKGROUND READING:

- Some knowledge of elementary number theory will be assumed, see, e.g., the Intute resources.
- F. Q. Gouvea,
*p-adic Numbers: An Introduction*, Springer, Berlin (2000). - A. Baker,
*An introduction to p-adic numbers*