This is not a complete answer but a possible path for the OP:
The entropy function $H(\mathbf{p})=-\sum^n_{j=1}p_j\log p_j$ satisfies $0\leq H(\mathbf{p})\leq \log n$
Define the map $T$ on the cube $(0,1)^n$ as
$$T\mathbf{p}=-\frac1{H(\mathbf{p})}(p_1\log p_1,\ldots, p_n\log p_n)$$
$T$ is in fact defined on $D_n=[0,1]^n\setminus V_n$ where $V_n$ is the set of vertices of $[0,1]^n$. Further, $T$ maps $D_n$ onto the simplex $S^*:=S\setminus\{\mathbf{e}_1,\ldots, \mathbf{e}_n\}$ ($\mathbf{e}_j(i)=\delta_{ij}$ for $1\leq i,j\leq n$).
The OP suggests that the median $\mathbf{p}^*=\frac{1}{n}(1,\ldots,1)$, a fixed point of $T$ on $(0,1)^n$, is an attractor for the dynamical system $\mathbf{p}_{m+1}=T\mathbf{p}_m$ where $\mathbf{p}_0$ is in $(0,1)^n$. Numerical evidence sustains this. (See R code below that simulates the system $\mathbf{p}_{n+1}=T\mathbf{p}_n$).
Denote by $T_i(\mathbf{p})=-\frac{p_i\log p_i}{H(\mathbf{p})}$. Then, $T$ is differentiable as a function on $(0,1)^n$ and the Jacobian matrix $T'(\mathbf{p})$ has entries given by
\begin{align}
\partial_jT_i &=-\frac{(1+\log p_j) p_i \log p_i}{H^2(\mathbf{p})}\\
\partial_iT_i&=-\frac{\big(1+\log p_i\big)\big(H(\mathbf{p})+p_i\log p_i\big)}{H^2(\mathbf{p})}
\end{align}
for $1\leq i,j\leq n$, $i\neq j$. It follows that for each $i$,
$$
\partial_jT_i(\mathbf{p})\left\{\begin{array}{lcr}
<0 & \text{if} & j\neq i, \,0<p_j<e^{-1}\\
>0 &\text{if} &j\neq i, \,e^{-1}<p_j<1\\
<0 &\text{if} & j=i, \,e^{-1}<p_i<1\\
>0 &\text{if} & j=i, \,0<p_i<e^{-1}
\end{array}
\right. \tag{*}\label{sign}
$$
At $\mathbf{p}^*$
\begin{align}
T'(\mathbf{p}^*)=\frac{1}{\log n}\begin{pmatrix}
(\log n -1)\frac{n-1}{n} & \frac{1-\log n}{n} & \frac{1-\log n}{n}& \ldots & \frac{1-\log n}{n}\\
\frac{1-\log n}{n} & (\log n -1)\frac{n-1}{n} & \frac{1-\log n}{n}&\ldots & \frac{1-\log n}{n}\\
\vdots &\vdots &\ddots & \ldots &\vdots\\
\frac{1-\log n}{n} & \frac{1-\log n}{n}&\frac{1-\log n}{n} &\ldots&
(\log n -1)\frac{n-1}{n}
\end{pmatrix}
\end{align}
It is clear that $\lambda_1=0$ is an eigenvalue with one dimensional Eigen space generated by $\mathbf{u}:=[1,\ldots,1]^\intercal$. There is another eigenvalue, $\lambda_2=\frac{\log n-1}{\log n}$ whose corresponding eigenspace $(n-1)$-dimensional space is generated by the vectors
$\mathbf{e}_1-\mathbf{e_j}$, $j=2,\ldots, n$. As $|\lambda_k|<1$, $k=1,2$, we conclude that indeed, $\mathbf{p}^*$ is a (local) atractor.
It remains to show that it isa global atractor for $T$ on $(0,1)^n$. This, perhaps, can be deduced from \eref{star} which at first sight imply that the orbit of any point $p_0$ in the (relative) interior of $S^*$ moves inwards, i.e. away from the boundary $\delta S^*$.
Along lower dimensional faces of the simplex $S^*$, there are other (hyperbolic) atractors given by the medians of the lower dimensional faces.
This code simulates the dynamical system $\mathbf{p}_{n+1}=T(\mathbf{p_n})$.
slog <- function(x){
ifelse(x==0,0,x*log(x))
}
slog <- Vectorize(slog,vectorize.args = 'x')
entropy <- function(p){
-sum(slog(p))
}
myTfun <- function(p){
H <- entropy(p)
return( -slog(p)/H)
}
set.seed(143)
library(mcmc)
p0 <- c(0.7153878, 0.01568887, 0.1950531, 0.07387031)
p0 <- rdirichlet(1,c(1,1,1,1))
Tp <- matrix(NA,nrow=30,ncol = length(p0))
Tp[1,] <- p0
for(n in 2:nrow(Tp)){
Tp[n,] <- myTfun(Tp[n-1,])
}
residual <- apply(Tp- rep(1/ncol(p0),ncol(p0)),1, function(x){max(abs(x))})
plot(1:nrow(Tp),residual, type = 'o', col = 'blue',
xlab = 'n')
Tp
