Dynamics on a finite uniform probability space

On $X=\{1,\ldots,n\}$ , $n\geq1$ , let $\Sigma=\mathcal P(X)$ and $\mu(E)=|E|/n$ . The measure-preserving self-maps are exactly the permutations $S_n$ . Such a map is ergodic exactly when it is one $n$ -cycle.

Proof and invariant functions

Preservation of each singleton's probability forces exactly one preimage of each point. If the disjoint cycles are $C_1,\ldots,C_r$ , invariant sets are precisely their unions. Fixed functions are constant on each cycle, so $\dim\ker(U_T-I)=r$ .

Function algebra and state

The algebra is $A=\mathbb C^n$ with coordinatewise multiplication, complex conjugation as involution, unit $(1,\ldots,1)$ , and maximum norm. Its state is $\varphi(z)=n^{-1}\sum_j z_j$ . Inverse pullback is

\alpha_T(z)_j=z_{T^{-1}(j)},\qquad A^{\alpha_T}\cong\mathbb C^r.

The vector-space dimension is $n$ ; this algebra is not the matrix algebra $M_n(\mathbb C)$ , whose dimension is $n^2$ .

Three points

For $T=(123)$ , $\alpha_T(z_1,z_2,z_3)=(z_3,z_1,z_2)$ , and only constant triples are fixed. For $T=(12)$ , fixed triples have form $(a,a,b)$ . The identity fixes every triple. The state of the event $\{1,3\}$ is $\varphi(1,0,1)=2/3$ .

Frequencies and averages

For $X=\mathbb Z/n\mathbb Z$ , $T(k)=k+1$ , the functions $e_j(k)=e^{2\pi ijk/n}$ form an orthonormal eigenbasis, with $U_Te_j=e^{2\pi ij/n}e_j$ . For $n>1$ , the system is ergodic but not weakly mixing. An orbit average converges to the average over the starting point's cycle.