The uniform distribution of sequences generated by iterated polynomials

Assume that $m,s\in \mathbb{N}$, $m>1$, while $f$ is a polynomial with integer coefficients, $\text{deg }f>1$, $f^{(i)}$ is the $i$th iteration of the polynomial $f$, ${\kappa }_n$ has a discrete uniform distribution on the set $\{0,1,\dots ,m^n-1\}$. We are going to prove that with $n$ tending to infinity random vectors $\left(\frac{\kappa_n}{m^n},\frac{f(\kappa_n) \bmod m^n}{m^n},\ldots,\frac{f^{(s - 1)}(\kappa_n) \bmod m^n}{m^n}\right)$ weakly converge to a vector having a continuous uniform distribution in the $s$-dimensional unit cube. Analogous results were obtained earlier only for some classes of polynomials with $s⩽3,\text{deg }f=2$. The mentioned vectors represent sequential pseudorandom numbers produced by a polynomial congruential generator modulo $m^n$.