English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We describe the set of all $(3,1)$-rational functions given on the set ofcomplex $p$-adic field $\mathbb C_p$ and having a unique fixed point. We study$p$-adic dynamical systems generated by such $(3,1)$-rational functions andshow that the fixed point is indifferent and therefore the convergence of thetrajectories is not the typical case for the dynamical systems. We obtainSiegel disks of these dynamical systems. Moreover an upper bound for the set oflimit points of each trajectory is given. For each $(3,1)$-rational function on$\mathbb C_p$ there is a point $\hat x=\hat x(f)\in \mathbb C_p$ which is zeroin its denominator. We give explicit formulas of radii of spheres (with thecenter at the fixed point) containing some points that the trajectories (underactions of $f$) of the points after a finite step come to $\hat x$. For a classof $(3,1)$-rational functions defined on the set of $p$-adic numbers $\mathbbQ_p$ we study ergodicity properties of the corresponding dynamical systems. Weshow that if $p\geq 3$ then the $p$-adic dynamical system reduced on eachinvariant sphere is not ergodic with respect to Haar measure. For $p=2$, undersome conditions we prove non ergodicity and show that there exists a sphere onwhich the dynamical system is ergodic. Finally, we give a characterization ofperiodic orbits and some uniformly local properties of the $(3.1)-$rationalfunctions.

$p$-adic dynamical systems of $(3,1)$-rational functions with unique fixed point