Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamical systems of the $p$-adic $(2,2)$-rational functions with two
  fixed points   We consider a family of $(2,2)$-rational functions given on the set of
complex $p$-adic field $\mathcal{C}_p$. Each such function $f$ has the two
distinct fixed points $x_1=x_1(f)$, $x_2=x_2(f)$. We study $p$-adic dynamical
systems generated by the $(2,2)$-rational functions. We prove that $x_1$ is
always indifferent fixed point for $f$, i.e., $x_1$ is a center of some Siegel
disk $SI(x_1)$. Depending on the parameters of the function $f$, the type of
the fixed point $x_2$ may be any possibility: indifferent, attractor, repeller.
We find Siegel disk or basin of attraction of the fixed point $x_2$, when $x_2$
is indifferent or attractor, respectively. When $x_2$ is repeller we find an
open ball any point of which repelled from $x_2$. Moreover, we study relations
between the sets $SI(x_1)$ and $SI(x_2)$ when $x_2$ is indifferent. For each
$(2,2)$-rational function on $\mathcal{C}_p$ there are two points $\hat
x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathcal{C}_p$ which are zeros of
its denominator. We give explicit formulas of radiuses of spheres (with the
center at the fixed point $x_1$) containing some points such that the
trajectories (under actions of $f$) of the points after a finite step come to
$\hat x_1$ or $\hat x_2$. We study periodic orbits of the dynamical system and
find an invariant set, which contains all periodic orbits. Moreover, we study
ergodicity properties of the dynamical system on each invariant sphere. Under
some conditions we show that the system is ergodic iff $p=2$.

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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex