We study number theoretic properties of the map x↦xx(modp), where x∈{1,2,…,p−1}, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p<N for which the map only has the trivial fixed point x=1. A key technical result, possibly of independent interest, is the existence of subsets Nq⊂{2,3,…,q−1} such that almost all k-tuples of distinct integers n1,n2,…,nk∈Nq are multiplicatively independent (if k is not too large), and |Nq|=q⋅(1+o(1)) as q→∞. For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of Fq-linear forms {Ln}n=2 q−1 “behaves randomly” in the sense that |{v∈Fq d:Ln(v)=1,n=2,3,…,q−1}|∼qd(1−1/q)q∼qd/e. (Here d=π(q−1) and the coefficients of Ln are given by the exponents in the prime power factorisation of n.)