---
title: Van der Put Theorem Overview
url: https://www.emergentmind.com/topics/van-der-put-theorem
type: topic
---

# Van der Put Theorem Overview

Searching arXiv for papers on van der Put expansions and related automata/dynamics results.
The classical van der Put theorem is the expansion theorem for continuous nonarchimedean functions: every continuous map \(f:\mathbb Z_p\to\mathbb Z_p\) admits a unique series expansion in characteristic functions of \(p\)-adic balls. Current arXiv usage suggests that the name also functions as a shorthand for several coefficient criteria built on that expansion, especially for \(1\)-Lipschitz maps, finite-state transducers, measure-preserving and ergodic \(p\)-adic dynamics, and—under the distinct phrase “Gerritzen–van der Put conjecture”—for a geometric comparison of fixed-point and branch-point configurations in split degenerate superelliptic curves [1112.5089] [1210.5925] [2407.11303].

## 1. Classical expansion theorem

For a continuous function \(f:\mathbb Z_p\to\mathbb Z_p\), the van der Put expansion has the form
\[
f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),
\]
where \(\chi(m,x)\) is the characteristic function of a basic \(p\)-adic ball. In the normalization used in the automata and dynamics papers, \(\chi(m,x)=1\) exactly when \(x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}\), with the usual convention at \(m=0\). The coefficients are uniquely determined by
\[
B_m=
\begin{cases}
f(m), & m<p,\\[2mm]
f(m)-f(m-m_{n-1}p^{n-1}), & m\ge p,
\end{cases}
\]
where \(m=m_0+m_1p+\cdots+m_{n-1}p^{n-1}\) and \(m_{n-1}\neq 0\). Thus \(B_m\) records the increment from the truncation that removes the highest nonzero base-\(p\) digit of \(m\) [1112.5089].

An analogous statement is used over \(\mathbb Z_d\) for arbitrary \(d\ge 2\). There the basis functions are cylinder indicators
\[
\chi_n(x)=\mathbf 1_{[n]_d X^\infty}(x),
\]
and every continuous \(f:\mathbb Z_d\to\mathbb Z_d\) has a unique expansion
\[
f(x)=\sum_{n\ge 0} B_n^f \chi_n(x),
\qquad
B_n^f=
\begin{cases}
f(n), & 0\le n<d,\\
f(n)-f(n_-), & n\ge d.
\end{cases}
\]
The \(d\)-adic/tree-theoretic formulation interprets \(\mathbb Z_d\) as the boundary of the rooted \(d\)-ary tree \(T_d\) [2006.02316].

The significance of the theorem is structural rather than merely representational. Because the basis functions are characteristic functions of ultrametric balls, the coefficients encode local behavior scale by scale. This ultrametric locality is exactly what makes the basis effective in automata theory and \(p\)-adic dynamics.

## 2. \(1\)-Lipschitz and compatible functions in van der Put form

A central refinement of the expansion theorem is the characterization of \(1\)-Lipschitz maps by divisibility properties of van der Put coefficients. In the prime-base setting, Anashin–Khrennikov–Yurova prove that
\[
f:\mathbb Z_p\to\mathbb Z_p \text{ is \(1\)-Lipschitz }
\iff
f(x)=\sum_{m=0}^{\infty} b_m\,p^{\lfloor \log_p m\rfloor}\chi(m,x),
\quad b_m\in\mathbb Z_p.
\]
Equivalently, the ordinary coefficient \(B_m\) is divisible by \(p^{\lfloor \log_p m\rfloor}\) for every \(m\) [1112.5089].

The same phenomenon holds for arbitrary \(d\ge 2\):
\[
f:\mathbb Z_d\to\mathbb Z_d \text{ is \(1\)-Lipschitz }
\iff
f(x)=\sum_{n\ge 0} b_n^f\, d^{\lfloor\log_d n\rfloor}\chi_n(x),
\quad b_n^f\in\mathbb Z_d.
\]
The coefficients \(b_n^f\) are the reduced van der Put coefficients [2006.02316].

In the \(p\)-adic dynamics literature, “compatible” is equivalent to \(1\)-Lipschitz. One formulation is
\[
|f(x)-f(y)|_p\le |x-y|_p,
\]
and another is preservation of congruences modulo all powers of \(p\). In van der Put form this becomes the coefficient growth condition
\[
B_m=p^{\lfloor \log_p m\rfloor} b_m,\qquad b_m\in\mathbb Z_p,
\]
or equivalently
\[
|B_m|_p\le p^{-\lfloor \log_p m\rfloor}.
\]
This coefficient criterion is repeatedly used as the basic compatibility test in dynamical applications [1210.5925].

The automata interpretation is exact: automaton functions are precisely the \(1\)-Lipschitz self-maps of \(\mathbb Z_p\). Moreover, the digitwise form
\[
f(x)=\sum_{i=0}^{\infty}\psi_i(x_0,\dots,x_i)p^i
\]
shows that the \(i\)-th output digit depends only on the first \(i+1\) input digits, never on higher digits [1112.5089]. This suggests that the van der Put basis is not only analytic but also intrinsically finite-prefix in character.

## 3. Finite-state automata criterion over \(\mathbb Z_p\)

The main theorem of the automata paper is a finiteness criterion stated entirely in terms of the normalized van der Put coefficients. Let
\[
f(x)=\sum_{m=0}^{\infty} b_m p^{\lfloor \log_p m\rfloor}\chi(m,x)
\]
be \(1\)-Lipschitz. Then \(f\) is the automaton function of a finite automaton if and only if both of the following hold:

1. the set \(\{b_m:m=0,1,2,\dots\}\) is a finite subset \(B_f\subset \mathbb Q\cap\mathbb Z_p\);
2. the \(p\)-kernel of \((b_m)_{m=0}^\infty\) is finite.

The \(p\)-kernel is
\[
\ker_p(a)=\left\{(a_{jp^m+t})_{j=0}^\infty : m=0,1,2,\dots,\ 0\le t<p^m\right\},
\]
and by the classical criterion a sequence is \(p\)-automatic if and only if its \(p\)-kernel is finite. Accordingly, the theorem may be restated as: a \(1\)-Lipschitz map comes from a finite-state \(p\)-ary transducer exactly when its normalized van der Put coefficients are finite-valued and \(p\)-automatic [1112.5089].

The paper also derives a Christol-type reformulation. If \(p\) is prime and \(B_f\) is embedded into a finite field \(\mathbb F_{p^\ell}\) by an injection \(\tau\), then finite-state realizability is equivalent to algebraicity of
\[
\sum_{m=0}^{\infty}\tau(b_m)X^m
\]
over \(\mathbb F_{p^\ell}(X)\) [1112.5089].

The proof proceeds by analyzing section functions
\[
f_{n,k}(z)=\frac{1}{p^k}\Bigl(f(n+p^k z)-\bigl(f(n)\bmod p^k\bigr)\Bigr),
\]
which correspond to states reached after reading a \(k\)-digit prefix. Finite-state behavior is equivalent to finiteness of the family \(\{f_{n,k}\}\). After decomposing \(f(n+p^k z)\) into a constant part and a tail part, the paper identifies the tail with the subsequences \((b_{n+p^k t})_{t\ge 1}\), so finiteness of sections becomes finiteness of the \(p\)-kernel. The rationality condition \(b_m\in\mathbb Q\cap\mathbb Z_p\) arises from eventual periodicity of the relevant \(p\)-adic constants [1112.5089].

A model example is the identity map \(f(x)=x\). Its normalized coefficients \(b_m\) are the leading base-\(p\) digit of \(m\), hence take values in \(\{0,1,\dots,p-1\}\) and form a \(p\)-automatic sequence. The theorem therefore recovers the obvious fact that \(x\mapsto x\) is realized by a one-state transducer that outputs each input digit unchanged [1112.5089].

## 4. Generalization to arbitrary \(d\) and the Mealy–Moore correspondence

The prime-base criterion was generalized from \(p\) to an arbitrary integer \(d\ge 2\) in the setting of rooted-tree endomorphisms and solenoid maps. If \(g\in\mathrm{End}(X^*)\) is an endomorphism of the rooted \(d\)-ary tree and \(\hat g:\mathbb Z_d\to\mathbb Z_d\) is the induced \(1\)-Lipschitz map, then \(g\) is finite state if and only if the sequence \((b_n^{\hat g})_{n\ge 1}\) of reduced van der Put coefficients satisfies two conditions:

1. it consists of finitely many eventually periodic elements of \(\mathbb Z_d\);
2. it is \(d\)-automatic.

For prime \(d=p\), this is explicitly presented as Anashin’s theorem; the contribution is its extension to all \(d\ge 2\) [2006.02316].

The paper makes the relation between coefficient sequences and automata explicit in both directions. Given a finite Mealy automaton defining \(g\), there is an explicit algorithmic procedure constructing a finite Moore automaton generating the sequence \((b_n^g)_{n\ge 0}\). Conversely, given a finite Moore automaton generating a sequence \((c_n)_{n\ge 0}\) of eventually periodic \(d\)-adic integers, there is an explicit algorithmic procedure constructing a finite Mealy automaton of an endomorphism \(g\) with \(b_n^g=c_n\) for all \(n\ge 0\). The two constructions are dual in the sense that the automata produced cover the input automata as labeled graphs [2006.02316].

The key bridge is the section formula. If \(g|_x\) denotes the section at \(x\in X\), then
\[
b_n^{g|_x}=
\begin{cases}
\sigma(b_x^g),& n=0,\\
b_{x+nd}^g+\sigma(b_x^g),& 0<n<d,\\
b_{x+nd}^g,& n\ge d,
\end{cases}
\]
where \(\sigma(a)=\dfrac{a-(a\bmod d)}{d}\). Deep coefficients of a section are therefore essentially a reindexing of the original coefficient sequence, with only a first-level correction term [2006.02316].

The examples are deliberately concrete. One example computes the reduced van der Put coefficients for a generator of the lamplighter group, producing eventually periodic \(2\)-adic coefficients such as \(b_0^p=1^\infty\), \(b_1^p=001^\infty\), \(b_2^p=101^\infty\), \(b_3^p=1^\infty\). Another starts from the Thue–Morse sequence viewed as \(2\)-adic values \(0^\infty\) and \(10^\infty\), prescribes it as \((b_n^t)_{n\ge 0}\), and constructs a \(2\)-state Mealy automaton with that coefficient sequence [2006.02316].

A significant limitation is also explicit: the Christol-type algebraicity argument used in the prime case does not obviously extend to general \(d\), and no such analogue is provided [2006.02316].

## 5. Measure preservation and ergodicity in \(p\)-adic dynamics

In \(p\)-adic dynamics, the van der Put basis yields coefficient criteria for Haar measure preservation and, in more specialized form, ergodicity. For a compatible function
\[
f(x)=\sum_{m=0}^{\infty} p^{\lfloor \log_p m\rfloor} b_m \chi(m,x),
\qquad b_m\in\mathbb Z_p,
\]
the measure-preserving criterion is exact: \(f\) preserves Haar measure if and only if \(b_0,b_1,\dots,b_{p-1}\) form a complete set of residues modulo \(p\), and for every \(k=1,2,3,\dots\) and every \(m=0,\dots,p^k-1\), the coefficients
\[
b_{m+p^k},\, b_{m+2p^k},\,\dots,\, b_{m+(p-1)p^k}
\]
are all nonzero residues modulo \(p\) [1210.5925].

This criterion is equivalent to bijectivity modulo \(p^k\) for all \(k\ge 1\). The proof uses the van der Put expansion to lift solutions from modulo \(p^k\) to modulo \(p^{k+1}\) one digit at a time. The same paper gives an additive normal form:
\[
f(x)=\xi(x)+p\cdot h(x),
\]
where \(h\) is arbitrary compatible and \(\xi(x)\) is assembled from permutations \(G\) of \(\{0,1,\dots,p-1\}\) and \(g_m\) of \(\{1,\dots,p-1\}\) through a van der Put series [1210.5925].

For ergodicity, the situation is more delicate. One paper provides sufficient conditions for general \(\mathbb Z_p\) and alternative proofs of the sharp \(\mathbb Z_2\) criteria. If
\[
f(x)=\sum_{m=0}^\infty B_m \chi(m,x)
\]
is \(1\)-Lipschitz, then
\[
|B_m|_p \le p^{-\lfloor \log_p m\rfloor}
\]
is the basic coefficient criterion for \(1\)-Lipschitzness. A sufficient measure-preserving condition is that \(\{B_0,\dots,B_{p-1}\}\) be distinct modulo \(p\) and
\[
B_m \equiv q(m)\pmod{p^{\lfloor \log_p m\rfloor+1}}
\qquad (m\ge p),
\]
where \(q(m)\) is the highest nonzero digit term of \(m\) [1210.5001].

The main general ergodicity result is a sufficient criterion: a \(1\)-Lipschitz function satisfying the coefficient conditions of Theorem 3.7 together with
\[
B_m\equiv B_0+m\pmod p,\qquad 0<m<p,
\]
is ergodic [1210.5001]. In the special case \(\mathbb Z_2\), the criterion is sharp. For
\[
f(x)=b_0\chi(0,x)+\sum_{m=1}^\infty 2^{\lfloor \log_2 m\rfloor} b_m \chi(m,x),
\qquad b_m\in\mathbb Z_2,
\]
ergodicity is equivalent to
\[
b_0\equiv 1\pmod 2,\qquad
b_0+b_1\equiv 3\pmod 4,\qquad
b_2+b_3\equiv 2\pmod 4,
\]
\[
|b_m|_2=1 \quad (m\ge 2),
\qquad
\sum_{m=2^{n-1}}^{2^n-1} b_m \equiv 0\pmod 4 \quad (n\ge 3)
\]
[1210.5001].

These dynamical results do not redefine the classical van der Put theorem; rather, they turn the basis into an exact coordinate system for local permutation data and scale-by-scale transitivity.

## 6. The separate geometric usage: the Gerritzen–van der Put conjecture

A distinct usage of “van der Put theorem” arises in nonarchimedean geometry through the Gerritzen–van der Put conjecture on split degenerate hyperelliptic and superelliptic curves. Here the object is not the van der Put basis but a comparison between two finite subsets of \(\mathbb P^1_K\): the fixed-point set
\[
S=\{a_0,b_0,\dots,a_g,b_g\}
\]
of order-\(p\) generators of a \(p\)-Whittaker group \(\Gamma_0\subset \mathrm{PGL}_2(K)\), and the branch set \(\mathcal B\) of the corresponding cyclic \(p\)-cover [2407.11303].

The original Gerritzen–van der Put statement asserted that \(S\) and \(\mathcal B\) have the same position. The 2024 paper shows that this literal formulation is false in general and requires a modification: \(S\) must be assumed optimal. Under that hypothesis, if \(\pi:S\to\mathcal B\) is the induced bijection, then for every subset \(\mathfrak s\subset S\) with \(2\le \#\mathfrak s\le 2g\),
\[
\mathfrak s \text{ is a cluster of }S
\iff
\pi(\mathfrak s)\text{ is a cluster of }\mathcal B.
\]
The theorem also gives depth transformations:
\[
\delta(\pi(\mathfrak s)) = p\,\delta(\mathfrak s)
\]
for odd-cardinality clusters,
\[
\delta(\pi(\mathfrak s)) = \delta(\mathfrak s)
\]
for even-cardinality clusters in the tame case, and
\[
\delta(\pi(\mathfrak s)) = \delta(\mathfrak s)+2v(p)
\]
for certain even clusters in the wild case [2407.11303].

The stronger statement is Berkovich-theoretic. If \(\Sigma_S\) and \(\Sigma_{\mathcal B}\) are the convex hulls of \(S\) and \(\mathcal B\) in the Berkovich projective line, then \(\pi\) extends to a homeomorphism
\[
\pi_*:\Sigma_S \stackrel{\sim}{\to} \Sigma_{\mathcal B}
\]
satisfying
\[
\delta(\pi_*(v),\pi_*(w)) = \delta(v,w)+(p-1)\mu(v,w),
\]
where \(\mu(v,w)\) measures the part of the path \([v,w]\) lying within distance \(\frac{v(p)}{p-1}\) of the fixed-point axes. In residue characteristic prime to \(p\), this acts by dilating each axis by factor \(p\) and leaving the rest unchanged; in residue characteristic \(p\), the dilation occurs on the radius-\(\frac{v(p)}{p-1}\) tubular neighborhood of each axis [2407.11303].

This geometric line of work is conceptually separate from the classical analytic van der Put theorem. The shared name comes from Gerritzen and van der Put, not from the basis theorem for continuous \(p\)-adic functions.

Source: https://www.emergentmind.com/topics/van-der-put-theorem