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Abstract Given a set$$S=\{x^2+c_1,\dots,x^2+c_s\}$$defined over a field and an infinite sequence$$\gamma$$of elements ofS, one can associate an arboreal representation to$$\gamma$$, generalising the case of iterating a single polynomial. We study the probability that a random sequence$$\gamma$$produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most setsSdefined over$$\mathbb{Z}[t]$$, and we conjecture a similar positive-probability result for suitable sets over$$\mathbb{Q}$$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify allSpossessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration. 

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families $${\mathcal{F}}$$ of polynomial maps, such as the family $$f_{c}(x)=x^{m}+c$$ , where $$m\geq 2$$ . We do this by making use of the dynatomic modular curves $$Y_{1}(n)$$ (respectively $$Y_{0}(n)$$ ) which parametrize maps $$f$$ in $${\mathcal{F}}$$ together with a point (respectively orbit) of period $$n$$ for $$f$$ . The key point in our strategy is to study the set of primes $$p$$ for which the reduction of $$Y_{1}(n)$$ modulo $$p$$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $$n$$ , a discriminant $$D_{n}$$ whose list of prime factors contains all the primes of bad reduction for $$Y_{1}(n)$$ . In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $$p$$ dividing $$D_{n}$$ : one guarantees that $$p$$ is in fact a prime of bad reduction for $$Y_{1}(n)$$ , yet this same criterion implies that $$Y_{0}(n)$$ is geometrically irreducible. The other guarantees that the reduction of $$Y_{1}(n)$$ modulo $$p$$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $$Y_{1}(n)$$ for several primes dividing $$D_{n}$$ when $n=7,8,11$ , and $$f_{c}(x)=x^{2}+c$$ . The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.