Van der Put Theorem Overview

• Van der Put Theorem is an expansion theorem that uniquely represents continuous p-adic functions using characteristic functions of ultrametric balls.
• It establishes coefficient criteria for 1-Lipschitz maps, linking analytic representations with automata theory and p-adic dynamics.
• The theorem also connects with geometric applications, such as the Gerritzen–van der Put conjecture, bridging fixed-point configurations and branch-point clusters.

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 (Anashin, 2011, Khrennikov et al., 2012, Yelton, 2024).

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 $p$0. The coefficients are uniquely determined by

$p$1

where $p$2 and $p$3. Thus $p$4 records the increment from the truncation that removes the highest nonzero base-$p$5 digit of $p$6 (Anashin, 2011).

An analogous statement is used over $p$7 for arbitrary $p$8. There the basis functions are cylinder indicators

$p$9

and every continuous $1$0 has a unique expansion

$1$1

The $1$2-adic/tree-theoretic formulation interprets $1$3 as the boundary of the rooted $1$4-ary tree $1$5 (Grigorchuk et al., 2020).

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 $1$6-adic dynamics.

2. $1$7-Lipschitz and compatible functions in van der Put form

A central refinement of the expansion theorem is the characterization of $1$8-Lipschitz maps by divisibility properties of van der Put coefficients. In the prime-base setting, Anashin–Khrennikov–Yurova prove that

$1$9

Equivalently, the ordinary coefficient $p$0 is divisible by $p$1 for every $p$2 (Anashin, 2011).

The same phenomenon holds for arbitrary $p$3: $p$4 The coefficients $p$5 are the reduced van der Put coefficients (Grigorchuk et al., 2020).

In the $p$6-adic dynamics literature, “compatible” is equivalent to $p$7-Lipschitz. One formulation is

$p$8

and another is preservation of congruences modulo all powers of $p$9. In van der Put form this becomes the coefficient growth condition

$f:\mathbb Z_p\to\mathbb Z_p$0

or equivalently

$f:\mathbb Z_p\to\mathbb Z_p$1

This coefficient criterion is repeatedly used as the basic compatibility test in dynamical applications (Khrennikov et al., 2012).

The automata interpretation is exact: automaton functions are precisely the $f:\mathbb Z_p\to\mathbb Z_p$2-Lipschitz self-maps of $f:\mathbb Z_p\to\mathbb Z_p$3. Moreover, the digitwise form

$f:\mathbb Z_p\to\mathbb Z_p$4

shows that the $f:\mathbb Z_p\to\mathbb Z_p$5-th output digit depends only on the first $f:\mathbb Z_p\to\mathbb Z_p$6 input digits, never on higher digits (Anashin, 2011). 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 $f:\mathbb Z_p\to\mathbb Z_p$7

The main theorem of the automata paper is a finiteness criterion stated entirely in terms of the normalized van der Put coefficients. Let

$f:\mathbb Z_p\to\mathbb Z_p$8

be $f:\mathbb Z_p\to\mathbb Z_p$9-Lipschitz. Then $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$0 is the automaton function of a finite automaton if and only if both of the following hold:

1. the set $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$1 is a finite subset $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$2;
2. the $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$3-kernel of $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$4 is finite.

The $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$5-kernel is

$f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$6

and by the classical criterion a sequence is $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$7-automatic if and only if its $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$8-kernel is finite. Accordingly, the theorem may be restated as: a $f(x)=\sum_{m=0}^{\infty} B_m\,\chi(m,x),$9-Lipschitz map comes from a finite-state $\chi(m,x)$0-ary transducer exactly when its normalized van der Put coefficients are finite-valued and $\chi(m,x)$1-automatic (Anashin, 2011).

The paper also derives a Christol-type reformulation. If $\chi(m,x)$2 is prime and $\chi(m,x)$3 is embedded into a finite field $\chi(m,x)$4 by an injection $\chi(m,x)$5, then finite-state realizability is equivalent to algebraicity of

$\chi(m,x)$6

over $\chi(m,x)$7 (Anashin, 2011).

The proof proceeds by analyzing section functions

$\chi(m,x)$8

which correspond to states reached after reading a $\chi(m,x)$9-digit prefix. Finite-state behavior is equivalent to finiteness of the family $p$0. After decomposing $p$1 into a constant part and a tail part, the paper identifies the tail with the subsequences $p$2, so finiteness of sections becomes finiteness of the $p$3-kernel. The rationality condition $p$4 arises from eventual periodicity of the relevant $p$5-adic constants (Anashin, 2011).

A model example is the identity map $p$6. Its normalized coefficients $p$7 are the leading base-$p$8 digit of $p$9, hence take values in $\chi(m,x)=1$0 and form a $\chi(m,x)=1$1-automatic sequence. The theorem therefore recovers the obvious fact that $\chi(m,x)=1$2 is realized by a one-state transducer that outputs each input digit unchanged (Anashin, 2011).

4. Generalization to arbitrary $\chi(m,x)=1$3 and the Mealy–Moore correspondence

The prime-base criterion was generalized from $\chi(m,x)=1$4 to an arbitrary integer $\chi(m,x)=1$5 in the setting of rooted-tree endomorphisms and solenoid maps. If $\chi(m,x)=1$6 is an endomorphism of the rooted $\chi(m,x)=1$7-ary tree and $\chi(m,x)=1$8 is the induced $\chi(m,x)=1$9-Lipschitz map, then $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$0 is finite state if and only if the sequence $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$1 of reduced van der Put coefficients satisfies two conditions:

1. it consists of finitely many eventually periodic elements of $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$2;
2. it is $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$3-automatic.

For prime $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$4, this is explicitly presented as Anashin’s theorem; the contribution is its extension to all $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$5 (Grigorchuk et al., 2020).

The paper makes the relation between coefficient sequences and automata explicit in both directions. Given a finite Mealy automaton defining $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$6, there is an explicit algorithmic procedure constructing a finite Moore automaton generating the sequence $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$7. Conversely, given a finite Moore automaton generating a sequence $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$8 of eventually periodic $x\equiv m \pmod{p^{\lfloor \log_p m\rfloor+1}}$9-adic integers, there is an explicit algorithmic procedure constructing a finite Mealy automaton of an endomorphism $p$00 with $p$01 for all $p$02. The two constructions are dual in the sense that the automata produced cover the input automata as labeled graphs (Grigorchuk et al., 2020).

The key bridge is the section formula. If $p$03 denotes the section at $p$04, then

$p$05

where $p$06. Deep coefficients of a section are therefore essentially a reindexing of the original coefficient sequence, with only a first-level correction term (Grigorchuk et al., 2020).

The examples are deliberately concrete. One example computes the reduced van der Put coefficients for a generator of the lamplighter group, producing eventually periodic $p$07-adic coefficients such as $p$08, $p$09, $p$10, $p$11. Another starts from the Thue–Morse sequence viewed as $p$12-adic values $p$13 and $p$14, prescribes it as $p$15, and constructs a $p$16-state Mealy automaton with that coefficient sequence (Grigorchuk et al., 2020).

A significant limitation is also explicit: the Christol-type algebraicity argument used in the prime case does not obviously extend to general $p$17, and no such analogue is provided (Grigorchuk et al., 2020).

5. Measure preservation and ergodicity in $p$18-adic dynamics

In $p$19-adic dynamics, the van der Put basis yields coefficient criteria for Haar measure preservation and, in more specialized form, ergodicity. For a compatible function

$p$20

the measure-preserving criterion is exact: $p$21 preserves Haar measure if and only if $p$22 form a complete set of residues modulo $p$23, and for every $p$24 and every $p$25, the coefficients

$p$26

are all nonzero residues modulo $p$27 (Khrennikov et al., 2012).

This criterion is equivalent to bijectivity modulo $p$28 for all $p$29. The proof uses the van der Put expansion to lift solutions from modulo $p$30 to modulo $p$31 one digit at a time. The same paper gives an additive normal form: $p$32 where $p$33 is arbitrary compatible and $p$34 is assembled from permutations $p$35 of $p$36 and $p$37 of $p$38 through a van der Put series (Khrennikov et al., 2012).

For ergodicity, the situation is more delicate. One paper provides sufficient conditions for general $p$39 and alternative proofs of the sharp $p$40 criteria. If

$p$41

is $p$42-Lipschitz, then

$p$43

is the basic coefficient criterion for $p$44-Lipschitzness. A sufficient measure-preserving condition is that $p$45 be distinct modulo $p$46 and

$p$47

where $p$48 is the highest nonzero digit term of $p$49 (Jeong, 2012).

The main general ergodicity result is a sufficient criterion: a $p$50-Lipschitz function satisfying the coefficient conditions of Theorem 3.7 together with

$p$51

is ergodic (Jeong, 2012). In the special case $p$52, the criterion is sharp. For

$p$53

ergodicity is equivalent to

$p$54

$p$55

(Jeong, 2012).

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 $p$56: the fixed-point set

$p$57

of order-$p$58 generators of a $p$59-Whittaker group $p$60, and the branch set $p$61 of the corresponding cyclic $p$62-cover (Yelton, 2024).

The original Gerritzen–van der Put statement asserted that $p$63 and $p$64 have the same position. The 2024 paper shows that this literal formulation is false in general and requires a modification: $p$65 must be assumed optimal. Under that hypothesis, if $p$66 is the induced bijection, then for every subset $p$67 with $p$68,

$p$69

The theorem also gives depth transformations: $p$70 for odd-cardinality clusters,

$p$71

for even-cardinality clusters in the tame case, and

$p$72

for certain even clusters in the wild case (Yelton, 2024).

The stronger statement is Berkovich-theoretic. If $p$73 and $p$74 are the convex hulls of $p$75 and $p$76 in the Berkovich projective line, then $p$77 extends to a homeomorphism

$p$78

satisfying

$p$79

where $p$80 measures the part of the path $p$81 lying within distance $p$82 of the fixed-point axes. In residue characteristic prime to $p$83, this acts by dilating each axis by factor $p$84 and leaving the rest unchanged; in residue characteristic $p$85, the dilation occurs on the radius-$p$86 tubular neighborhood of each axis (Yelton, 2024).

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$87-adic functions.