How can I write this type of convergent notation in LaTeX?

You can construct this symbol using a mixture of \angle (from amssymb) and \longrightarrow together with a combination of \ooalign and \mathpalette.

\documentclass{article}

\usepackage{amssymb}

\makeatletter
\newcommand{\angleto}{\mathpalette\angleto@\relax}
\newcommand{\angleto@}[2]{\mathrel{\ooalign{$#1\hidewidth\angle\hidewidth$\cr$#1\longrightarrow$}}}
\makeatother

\begin{document}

$z \angleto \zeta_{z \angleto \zeta_{z \angleto \zeta}}$

\end{document}

Like that:

\documentclass{article}
\usepackage{mathtools}


\begin{document}
    
    $z\mathrel{\rlap{\;\angle}{\longrightarrow}}\zeta\lowercase{o}$

\end{document}

A sixty degree angle:

\documentclass{article}
\usepackage{amsmath,pict2e}

\makeatletter
\newcommand{\angledto}{\mathrel{\mathpalette\angled@arrow\longrightarrow}}
\newcommand{\angledgets}{\mathrel{\mathpalette\angled@arrow\longleftarrow}}

\newcommand{\angled@arrow}[2]{%
  \vphantom{#1\rightarrow}%
  \ooalign{\hidewidth\angled@angle{#1}\hidewidth\cr$\m@th#1#2$\cr}%
}
\newcommand{\angled@angle}[1]{%
  $\m@th#1\vcenter{\hbox{%
    \sbox\z@{$#1\mathstrut$}%
    \setlength{\unitlength}{0.5\dimexpr\ht\z@+\dp\z@}% adjust the factor
    \begin{picture}(1.1547,1)
    \roundcap
    \roundjoin
    \Line(0.57735,1)(0,0)(1.1547,0)
    \end{picture}%
  }}$%
}
\makeatother

\begin{document}

\[
z\angledto \zeta_0
\]
\[
\scriptstyle z\angledto \zeta_0
\]
\[
\scriptscriptstyle z\angledto \zeta_0
\]

\end{document}

Here's with \angledgets

It wouldn't be difficult to reverse the angle, if requested.