Dynamical systems and ergodic theory Definition

P-adic dynamics

Builds on p-adic number Non-Archimedean absolute value P-adic analysis Iterated function

Definition

Let $p$ be a prime. Let $K$ be a field that is complete for a non-Archimedean absolute value $\abs{\cdot}$ extending the $p$ -adic absolute value of $\Q$ . Examples are $\Q_p$ , its finite extensions, and $\C_p$ . A $p$ -adic dynamical system is a pair $(X, \varphi)$ in which:

$X$ is either the projective line $\mathbb{P}^1(K)$ or a subset of $K$ , such as a closed disc;
$\varphi \colon X \to X$ is given by a rational function in $K(z)$ when $X = \mathbb{P}^1(K)$ , or by a power series with coefficients in $K$ that converges on $X$ otherwise.

$p$ -adic dynamics is the study of the iterates $\varphi^n$ and of the orbits $\set{\varphi^n(x) : n \in \N}$ of such systems, with respect to the topology of $\abs{\cdot}$ .

Remarks

Example: let $X = \Z_p$ and $\varphi(x) = x + 1$ . It maps $\Z_p$ into $\Z_p$ , and it is a polynomial, so $(\Z_p, \varphi)$ is a $p$ -adic dynamical system. It is an isometry, since $\abs{\varphi(x) - \varphi(y)} = \abs{x - y}$ . The orbit of $0$ is $\N$ , which is dense in $\Z_p$ .

Non-example: let $\varphi(x) = x / p$ on $X = \Z_p$ . Then $\varphi(1) = 1/p$ has absolute value $p > 1$ , so $\varphi$ does not map $X$ into $X$ , and clause 2 fails.

A second example: $\varphi(z) = z^2$ on the open unit disc $\set{x \in \C_p : \abs{x} < 1}$ . Since $\abs{\varphi^n(x)} = \abs{x}^{2^n}$ , every orbit tends to the attracting fixed point $0$ .

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Tag	`J1SGNEWKWS`
URI	`m2://J1SGNEWKWS`
Kind	definition
Writing	W0 stub
MSC	37P20, 11S82
Defines	P-adic dynamics
Uses	p-adic number, Non-Archimedean absolute value, P-adic analysis, Iterated function

P-adic dynamics

Definition

Remarks

Reading path

Further reading

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