  Big data, encompassing extensive datasets, has seen rapid expansion, notably
with a considerable portion being textual data, including strings and texts.
Simple compression methods and standard data structures prove inadequate for
processing these datasets, as they require decompression for usage or consume
extensive memory resources. Consequently, this motivation has led to the
development of compressed data structures that support various queries for a
given string, typically operating in polylogarithmic time and utilizing
compressed space proportional to the string's length. Notably, the suffix array
(SA) query is a critical component in implementing a suffix tree, which has a
broad spectrum of applications.
  A line of research has been conducted on (especially, static) compressed data
structures that support the SA query. A common finding from most of the studies
is the suboptimal space efficiency of existing compressed data structures.
Kociumaka, Navarro, and Prezza [IEEE Trans. Inf. Theory 2023] have made a
significant contribution by introducing an asymptotically minimal space
requirement, $O\left(\delta \log\frac{n\log\sigma}{\delta\log n} \log n
\right)$ bits ($\delta$-optimal space), sufficient to represent any string of
length $n$, with an alphabet size of $\sigma$, and substring complexity
$\delta$, serving as a measure of repetitiveness. More recently, Kempa and
Kociumaka [FOCS 2023] presented $\delta$-SA, a compressed data structure
supporting SA queries in $\delta$-optimal space. However, the data structures
introduced thus far are static.
  We present the first dynamic compressed data structure that supports the SA
query and update in polylogarithmic time and $\delta$-optimal space. More
precisely, it can answer SA queries and perform updates in $O(\log^7 n)$ and
expected $O(\log^8 n)$ time, respectively, using an expected $\delta$-optimal
space.
