Self-Power Escort Dynamics on the Probability Simplex

For a fixed integer \( n\ge2 \), let \( \Delta_{n-1} \) denote the standard probability simplex in $\( \mathbb R^n \)$. We introduce and analyze the self-power escort transformation \( T_n:\Delta_{n-1}\to\Delta_{n-1} \) defined by \( \begin{equation*}T_n(p)_i=\frac{p_i^{p_i}}{\sum_{j=1}^n p_j^{p_j}},\qquad i=1,\ldots,n,\end{equation*} \) with the continuous boundary convention \( 0^0=1 \). Unlike the usual power escort transformation, whose exponent is an external parameter, the exponent here is the coordinate itself; equivalently, each coordinate is reweighted by \( \exp(p_i\log p_i) \). The paper proves that this elementary self-feedback rule has a rigid global dynamics. Every boundary point is immediately activated, every orbit enters an explicitly described compact core after one step, the uniform distribution is the unique fixed point, and the Hilbert log-diameter contracts exponentially after core entrance. Consequently every orbit converges exponentially to the uniform distribution, and no nontrivial periodic orbit exists. We then compute the complete linear spectrum and second-order normal form at the uniform state, obtaining a sign transition between the binary and higher-dimensional systems. For \( n\ge4 \), the post-core Hilbert contraction constant is sharpened to a two-variable min--max problem involving only the smallest and largest coordinates. We also classify the entire one-step uniformization fiber \( T_n^{-1}(\nu_n) \): besides the vertices and the uniform point, each dimension \( n\ge3 \) has exactly one nonuniform interior orbit type, up to permutation. Finally, in the binary case we prove global contraction in logit coordinates and an arithmetic escape phenomenon: algebraic irrational inputs become transcendental after one step, while rational nonuniform inputs become transcendental after two steps.