On the proportion of irreducible polynomials in unicritically generated semigroups

Let p be a prime number and let $S=\{x^p+c_1,\dots ,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots ,c_r\in Z$. Moreover, assume that S contains at least one irreducible polynomial over $Q$. Then we construct a large, explicit subset of irreducible polynomials within the semigroup generated by S under composition; in fact, we show that this subset has positive asymptotic density within the full semigroup when we count polynomials by degree. In addition, when $p=2$ we construct an infinite family of semigroups that break the local-global principle for irreducibility. To do this, we use a mix of algebraic and arithmetic techniques and results, including Runge's method, the elliptic curve Chabauty method, and Fermat's Last Theorem.