# Identification of i.i.d. series

I have a time series

$$X(t+1)= X(t)$$

or

$$X(t+1)= 1-X(t)$$

with equal probability and $X$ has a uniform$(0,1)$ distribution. Is the series i.i.d.?

Your first observation in the time series, $X(0)$, is distributed as $X(0)\sim Unif(0, 1)$, and by the recursive definition of the time series all subsequent observations $X(t)$ for $t \geq 1$ can only take two values -- $X(0)$ with probability 0.5 and $1-X(0)$ with probability 0.5. As a result, $X(t)\sim Unif(0, 1)$ for all $t$, meaning all observations in the time series are identically distributed.
However, no pair of observations $X(t_1)$ and $X(t_2)$ in the time series ($t_1\neq t_2$) are independent of one another -- given that $X(t_1) = c$ for some constant $c$, $X(t_2)$ takes value $c$ with probability 0.5 and $1-c$ with probability 0.5.
It should be noted that conditioned on $X(0)$ taking some value $c$, all subsequent random variables $X(t)$ for $t \geq 1$ are IID discrete random variables taking value $c$ with probability 0.5 and $1-c$ with probability 0.5; whether $X(t_1)$ takes value $c$ or $1-c$ has no impact on whether $X(t_2)$ takes value $c$ or $1-c$ when $t_1\neq t_2$