# Questions tagged [arithmetic-dynamics]

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### Special value of the Artin--Mazur zeta function in arithmetic dynamics

Let $X$ be a compact manifold and $f: X\rightarrow X$ be a diffeomorphism. Assume that the $k$-fold iterate $f^k: X\rightarrow X$ has finitely many fixed points for all natural numbers $k$. The ...
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### Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...
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### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...
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### Algebraic Dynamics over separated schemes

I have a few questions regarding the current status of research on algebraic dynamics over separated schemes. In what follows $\varphi:X\rightarrow X$ will be a finite self-morphism of a noetherian ...
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### A follow up question related to entropy

For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
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### Greatest common divisor of a^{2^n}-1 and b^{2^n}-1

Let a and b be coprime integers. Do we know, expect, or unexpect that there are infinitely many primes p which divide $gcd(a^{2^n} - 1, b^{2^n}-1)$ for some n? Certainly any Fermat prime will ...
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### Rational maps whose complex conjugate equals a PGL conjugate

Let $f(z)\in\mathbb{C}(z)$ be a rational function, and let $\bar{f}(z)$ denote the function obtained by taking the complex conjugate of the coefficients of $f$. I am interested in maps $f$ for which ...
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### If you were to axiomatize the notion of entropy

What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...
Let $p$ be a prime, and consider the sequence $x_0, x_1, \dots$ of elements of the finite field $\mathbf F_p$ given by $x_0 = 0$ and $x_{i+1} = x_i^2 + 1$ for all $i \ge 0$. This sequence must ...