Questions tagged [arithmetic-dynamics]
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15
questions with no upvoted or accepted answers
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Strange formula in arithmetic dynamic
Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two.
We discovered the following operator which acts on the space of polynomials (or ...
13
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0
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548
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Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
12
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391
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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 ...
9
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311
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Discriminants of Gleason's period-$n$ polynomials for the Mandelbrot set
Gleason's polynomials are the sequence of monic integer polynomials defined recursively by
$$
\prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}],
$$
for ...
5
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215
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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 ...
4
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319
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Why are critical points important for dynamical systems?
I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (...
4
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145
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The image of annuli of the non-Archimedean projective line by rational functions
I'm reading the book "potential theory and dynamics over the Berkovich projective line" by Baker and Rumely. The proposition 2.18 in this claims that if you choose suitable finite $\{a_i\} \in D(a,r)$ ...
4
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328
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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 ...
3
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181
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Stronger estimates for dynamical analogue of a sum of multiplicative order and primes
Let $a$ be a positive integer and $\mathrm{ord}_{p}(a)$ denote the multiplicative order of $a$ modulo $p.$ We know by a result of Murty, Silverman and Rosen that the sum $$\sum_{p~\text{prime}} \frac{\...
3
votes
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290
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Paper by Moser on commuting circle diffeomorphisms and simultaneous Diophantine approximations
I am reading Moser's paper On commuting circle mappings and simultaneous Diophantine approximations and I found it hard because it is my first time that I seriously have to read a paper. It is a local ...
3
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132
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Equidistribution of double coset
Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...
2
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477
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Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
2
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128
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Can this construction generate bounded aperiodic functions?
This question is based on this old MathOverflow question: How this set of functions is ordered?
In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
1
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237
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A probability application question
Suppose there are two possible states $H$ and $L$, with prior probability $p$ and
$1-p$ respectively. There are infinite rounds with a discount factor $ d$. In
round 1, you could choose a value $t_1$...
0
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146
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cat map re-transformation
Hi,
Is there any way of moving from one cat map transformation to the other without resetting parameters?
For example, suppose you have two matrices '$A$'and '$B$' each permuted with different cat ...