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Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $\chi$ be the quadratic character of $\mathbb{F}_q$, and $\mathcal{O}_f(a)$ the forward orbit of $a$ under iteration by $f$. Suppose that the sequence $(\chi(f^n(a)))_{n\geq 1}$ is periodic, and $m$ is its period. Assuming a mild and generic condition on $f$, we show that, up to a constant, $m$ can be bounded from below by $|\mathcal{O}_f(a)|/q^\frac{2\log_{2}(d)+1}{2\log_2(d)+2}$. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant, we cannot have more than $q^\frac{2\log_2(d)+1}{2\log_2(d)+2}$ consecutive squares or non-squares in the forward orbit of $a$. In addition, we provide a classification of all polynomials for which our generic condition does not hold., Comment: 17 pages

In his earlier work, the author introduced a group theory question that arises in the study of iterated Galois groups of post-critically finite quadratic polynomials. In this paper, we prove the first non-trivial results on this question., Comment: To appear in Groups, Geometry, and Dynamics

Let $f$ be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of $f$, and conjectured that its limiting distribution explains the factorization of large iterates of $f$. Later on, Xia, Boston, and the author did extensive Magma computations and found some exceptional families of quadratics that do not seem to follow the original Markov model conjectured by Boston and Jones. They did this by empirically observing that certain factorization patterns predicted by the Boston-Jones model never seem to occur for these polynomials, and suggested a multi-step Markov model which takes these missing factorization patterns into account. In this note, we provide proofs for all these missing factorization patterns. These are the first provable results that explain why the original conjecture of Boston and Jones does not hold for all monic quadratic polynomials., Comment: 13 pages

Fix an integer $d\geq 2$. The parameters $c_0\in \bar{\mathbb{Q}}$ for which the unicritical polynomial $f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$ has finite postcritical orbit, also known as Misiurewicz parameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work in a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we call $p$-special, which may be of independent number theoretic interest., Comment: 23 pages

Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz points $c_0,c_1\in \bar{\mathbb{Q}}$ are such that the polynomials $f_{c_0,d}$ and $f_{c_1,d}$ have the same orbit type. One classical question is whether $c_0$ and $c_1$ need to be Galois conjugates or not. Recently there has been a partial progress on this question by several authors. In this note, we prove some new results when $d$ is a prime. All the results known so far were in the cases of period size at most $3$. In particular, our work is the first to say something provable in the cases of period size greater than $3$., Comment: 10 pages

For any $n\geq 1$, let $T_n$ be the complete binary rooted tree of height $n$, and $f(x)=(x+a)^2-a-1$ such that $a\neq \pm b^2$ for any $b\in \mathbb{Z}$. In \cite{Settled}, Jones and Boston empirically observed that iteratively applying a certain Markov process on the factorization types of $f$ gives rise to certain permutation groups $M_n(f)\leq \text{Aut}(T_n)$ for $n\leq 5$. We prove a refined version of this phenomenon for all $n$, and for all the irreducible post-critically finite quadratic polynomials with integer coefficients, except for certain conjugates of $x^2-2$. We do this by constructing these groups explicitly. Although there have already been some conjectures relating the Markov processes to the dynamics of quadratic polynomials, our results are the first to prove such a connection. If $f(x)\in \mathbb{Z}[x]$ is a post-critically finite quadratic polynomial, and $G_n(f)$ is the Galois group of $f^n$ over $\mathbb{Q}(i)$, then we conjecture that for all $n\geq 1$, $M_n(f)$ contains a subgroup isomorphic to $G_n(f)$, analogous to the role of Mumford-Tate groups in the classical arithmetic geometry. We provide evidence that this is implied by a purely group theoretical statement., Comment: 46 pages

In this paper, we study the critical orbit of a post-critically finite polynomial of the form $f_{c,d}(x) = x^d+c \in \mathbb{C}[x]$. We discover that in many cases the orbit elements satisfy some strong arithmetic properties. It is well known that the $c$ values for which $f_{c,d}$ has tail size $m\geq 1$ and period $n$ are the roots of a polynomial $G_d(m,n) \in \mathbb{Z}[x]$, and the irreducibility or not of $G_d(m,n)$ has been a great mystery. As a consequence of our work, for any prime $d$, we establish the irreducibility of these $G_d(m,n)$ polynomials for infinitely many pairs $(m,n)$. These appear to be the first known such infinite families of $(m,n)$. We also prove that all the iterates of $f_{c,d}$ are irreducible over $\mathbb{Q}(c)$ if $d$ is a prime and $f_{c,d}$ has a fixed point in its post-critical orbit., Comment: Final version of the article. To appear in Functiones et Approximatio Commentarii Mathematici

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over $\mathbb{Q}$. In particular, we obtain infinitely many new examples of the phenomenon studied in \cite{Jones}. While doing this, we also find, up to three possible exceptions, all integers $a$ such that all iterates of the quadratic polynomial $(x+a)^2-a-1$ are irreducible over $\mathbb{Q}$, which answers a question posed in \cite{AyadMcdonald}, except for three values of $a$. Finally, we make a conjecture that suggests a necessary and sufficient condition for the stability of any monic, post-critically finite quadratic polynomial over any field of characteristic $\neq 2$., Comment: Final version of the article. To appear in the Rocky Mountain Journal of Mathematics

Jones and Boston conjectured that the factorization process for iterates of irreducible quadratic polynomials over finite fields is approximated by a Markov model. In this paper, we find unexpected and intricate behavior for some quadratic polynomials, in particular for the ones with tail size one. We also propose a multi-step Markov model that explains these new observations better than the model of Jones and Boston., Comment: 13 pages