Schneider Continued Fraction Map

• The Schneider continued fraction map is a p-adic algorithm that expands numbers in Qₚ using valuation and residue digits, serving as a non-Archimedean analogue to the Gauss map.
• Its dynamics are modeled via symbolic coding with countable shifts, enabling explicit calculations of geometric potentials, pressure functions, and multifractal spectra.
• Recent work extends its framework to random continued fraction systems in real dynamics, blending Gauss and Rényi maps to generate semi-regular, sign-alternating expansions.

Searching arXiv for Schneider continued fraction map and related papers. arXiv search query: "Schneider continued fraction map" The Schneider continued fraction map denotes a $p$-adic continued-fraction algorithm whose dynamics generate continued fraction expansions for points in $\mathbb{Q}_p$, most commonly on the invariant ball $p\mathbb{Z}_p$. In recent work it is treated as a non-Archimedean analogue of the Gauss map, with explicit shift models, equilibrium states, Lyapunov spectra, and multifractal formulae (Liu, 2023, Alvarado et al., 9 Jan 2026, Alvarado et al., 6 May 2026). A related but distinct usage appears in the real one-dimensional random-dynamics literature, where a Bernoulli mixture of the Gauss and Rényi maps is described as a Schneider-type random continued fraction transformation generating semi-regular continued fractions (Kalle et al., 2015).

1. Terminological scope and variants

In the $p$-adic literature, Schneider’s continued fraction map is the map $T:\mathbb{Q}_p\to\mathbb{Q}_p$ defined by

$$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$

with $\varepsilon(0)=1$, where $[x]_p\in\{0,\dots,p-1\}$ is the unique digit satisfying $|x-[x]_p|_p<1$ for $x\in\mathbb{Z}_p$ (Liu, 2023). In the more specialized thermodynamic and multifractal studies, the same algorithm is restricted to

$\mathbb{Q}_p$0

and written in terms of the valuation $\mathbb{Q}_p$1 and a residue digit $\mathbb{Q}_p$2 (Alvarado et al., 9 Jan 2026, Alvarado et al., 6 May 2026).

A second usage arises in the real setting. The source description for the random continued fraction transformation emphasizes that the term “Schneider continued fraction map” is not explicitly used there, but that the system is naturally understood as a Schneider-type random map: at each step one chooses between the regular Gauss map and the backwards Rényi map, producing semi-regular continued fractions with signs $\mathbb{Q}_p$3 (Kalle et al., 2015). This suggests that the term has become partly contextual: in $\mathbb{Q}_p$4-adic dynamics it names a specific map, while in real random dynamics it can denote a family of semi-regular, sign-changing continued fraction algorithms of Schneider–Perron type.

2. $\mathbb{Q}_p$5-adic definition and continued fraction expansion

Let $\mathbb{Q}_p$6 be a prime, $\mathbb{Q}_p$7 the field of $\mathbb{Q}_p$8-adic numbers, $\mathbb{Q}_p$9, $p\mathbb{Z}_p$0, and

$p\mathbb{Z}_p$1

For $p\mathbb{Z}_p$2, one sets $p\mathbb{Z}_p$3. For $p\mathbb{Z}_p$4, the first coefficients are

$p\mathbb{Z}_p$5

and

$p\mathbb{Z}_p$6

The map is then

$p\mathbb{Z}_p$7

and one checks that $p\mathbb{Z}_p$8, so $p\mathbb{Z}_p$9 is well defined on $p$0 (Alvarado et al., 6 May 2026).

Higher coefficients are defined recursively by

$p$1

For points whose orbit never hits $p$2, this yields an infinite Schneider continued fraction; for each finite truncation level $p$3,

$p$4

The set

$p$5

is precisely the set of elements of $p$6 with finite Schneider continued fraction expansion (Alvarado et al., 6 May 2026).

The analogy with the real Gauss map is explicit in the cited work. In the real case, $p$7 yields digits $p$8. In the Schneider setting, the role of Euclidean division is replaced by the valuation and the residue class modulo $p$9: the exponents $T:\mathbb{Q}_p\to\mathbb{Q}_p$0 appear as powers of $T:\mathbb{Q}_p\to\mathbb{Q}_p$1 in the numerators, while the digits $T:\mathbb{Q}_p\to\mathbb{Q}_p$2 appear in the denominators (Liu, 2023, Alvarado et al., 9 Jan 2026).

3. Coding, shift models, and arithmetic characterization

One symbolic model uses the countable alphabet

$T:\mathbb{Q}_p\to\mathbb{Q}_p$3

with left shift $T:\mathbb{Q}_p\to\mathbb{Q}_p$4. If

$T:\mathbb{Q}_p\to\mathbb{Q}_p$5

then higher-order cylinders are

$T:\mathbb{Q}_p\to\mathbb{Q}_p$6

The coding map

$T:\mathbb{Q}_p\to\mathbb{Q}_p$7

is a homeomorphism, and

$T:\mathbb{Q}_p\to\mathbb{Q}_p$8

so $T:\mathbb{Q}_p\to\mathbb{Q}_p$9 is topologically conjugate to the full countable shift $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$0 (Alvarado et al., 9 Jan 2026).

A second coding is intrinsic to $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$1 itself. Define

$$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$2

and

$$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$3

Then

$$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$4

and $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$5 is a bijective isometry of $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$6; hence it is a topological conjugacy between $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$7 and $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$8 (Liu, 2023).

This conjugacy yields a sharp arithmetic criterion. The map $$T(x)=\frac{\varepsilon(x)}{x}-\Bigl[\frac{\varepsilon(x)}{x}\Bigr]_p,$$9 satisfies

$\varepsilon(0)=1$0

and for $\varepsilon(0)=1$1,

$\varepsilon(0)=1$2

Moreover, if $\varepsilon(0)=1$3, then there exists $\varepsilon(0)=1$4 such that

$\varepsilon(0)=1$5

The points $\varepsilon(0)=1$6 and $\varepsilon(0)=1$7 are fixed points of $\varepsilon(0)=1$8, with $\varepsilon(0)=1$9 and $[x]_p\in\{0,\dots,p-1\}$0 (Liu, 2023). In this respect, rationality is characterized dynamically through eventual entrance into a terminal state, paralleling the finiteness of real continued fractions.

4. Local geometry and thermodynamic formalism on $[x]_p\in\{0,\dots,p-1\}$1

On a basic cylinder $[x]_p\in\{0,\dots,p-1\}$2, the Schneider map has the explicit form

$[x]_p\in\{0,\dots,p-1\}$3

For $[x]_p\in\{0,\dots,p-1\}$4,

$[x]_p\in\{0,\dots,p-1\}$5

Thus $[x]_p\in\{0,\dots,p-1\}$6 is locally expanding on each cylinder with factor $[x]_p\in\{0,\dots,p-1\}$7 (Alvarado et al., 9 Jan 2026).

The diameter of $[x]_p\in\{0,\dots,p-1\}$8 is $[x]_p\in\{0,\dots,p-1\}$9, and this leads to the geometric potential

$|x-[x]_p|_p<1$0

Because $|x-[x]_p|_p<1$1 depends only on the first digit $|x-[x]_p|_p<1$2, it is locally constant on natural clopen partition elements, a feature that is central to the explicit calculations in the $|x-[x]_p|_p<1$3-adic theory (Alvarado et al., 9 Jan 2026, Alvarado et al., 6 May 2026).

For the potential $|x-[x]_p|_p<1$4, the pressure is

$|x-[x]_p|_p<1$5

For each $|x-[x]_p|_p<1$6, there exists a unique equilibrium state $|x-[x]_p|_p<1$7 for $|x-[x]_p|_p<1$8 and $|x-[x]_p|_p<1$9. In particular,

$x\in\mathbb{Z}_p$0

and the corresponding equilibrium state is the normalized Haar measure $x\in\mathbb{Z}_p$1, for which

$x\in\mathbb{Z}_p$2

The cited work also introduces bounded-digit subsystems

$x\in\mathbb{Z}_p$3

with restricted pressure

$x\in\mathbb{Z}_p$4

providing compact approximations for the full countable-alphabet system (Alvarado et al., 9 Jan 2026).

5. Lyapunov exponents, rational approximation, and multifractal power means

The Lyapunov exponent adapted to the non-Archimedean setting is

$x\in\mathbb{Z}_p$5

whenever the limit exists. Thus $x\in\mathbb{Z}_p$6 is $x\in\mathbb{Z}_p$7 times the asymptotic arithmetic mean of the Schneider digits $x\in\mathbb{Z}_p$8 (Alvarado et al., 9 Jan 2026).

For

$x\in\mathbb{Z}_p$9

the spectrum is given by

$\mathbb{Q}_p$00

Equivalently,

$\mathbb{Q}_p$01

and the infimum is attained at the unique

$\mathbb{Q}_p$02

The spectrum is real analytic on $\mathbb{Q}_p$03, and the typical exponent under Haar measure is

$\mathbb{Q}_p$04

(Alvarado et al., 9 Jan 2026).

The same Lyapunov exponent governs the rate of rational approximation by Schneider convergents. If $\mathbb{Q}_p$05 is the $\mathbb{Q}_p$06-th truncated Schneider convergent, then

$\mathbb{Q}_p$07

whenever the limit exists. The multifractal spectrum of approximation exponents is therefore identical to the Lyapunov spectrum (Alvarado et al., 9 Jan 2026).

A further development studies asymptotic power means

$\mathbb{Q}_p$08

and

$\mathbb{Q}_p$09

For Haar measure,

$\mathbb{Q}_p$10

and

$\mathbb{Q}_p$11

For the level sets

$\mathbb{Q}_p$12

there exists a unique $\mathbb{Q}_p$13 such that

$\mathbb{Q}_p$14

The parameter $\mathbb{Q}_p$15 is characterized by polylogarithmic equations: for $\mathbb{Q}_p$16,

$\mathbb{Q}_p$17

and for $\mathbb{Q}_p$18,

$\mathbb{Q}_p$19

The papers attribute the explicitness of these formulae to the locally constant character of the geometric potential, in sharp contrast with the classical real setting (Alvarado et al., 6 May 2026).

6. Random Schneider-type maps on the real interval

The real random continued fraction transformation studied in (Kalle et al., 2015) operates in Perron’s class of semi-regular continued fractions. Every $\mathbb{Q}_p$20 can be written as

$\mathbb{Q}_p$21

Regular continued fractions correspond to all $\mathbb{Q}_p$22, and Rényi’s backwards continued fractions to all $\mathbb{Q}_p$23 (Kalle et al., 2015).

The two basic interval maps are the Gauss map

$\mathbb{Q}_p$24

and the Rényi map

$\mathbb{Q}_p$25

The associated random skew product is

$\mathbb{Q}_p$26

and the continued-fraction coder $\mathbb{Q}_p$27 on $\mathbb{Q}_p$28 produces digits $\mathbb{Q}_p$29 and signs $\mathbb{Q}_p$30 (Kalle et al., 2015).

In the source description, this system is identified as a Schneider-type random continued fraction map: a Bernoulli mixture of Gauss and backwards maps generating semi-regular continued fractions. For irrational $\mathbb{Q}_p$31, every $\mathbb{Q}_p$32 yields a different expansion, so every irrational $\mathbb{Q}_p$33 has uncountably many such expansions. For rational $\mathbb{Q}_p$34, there are countably many expansions produced by $\mathbb{Q}_p$35 (Kalle et al., 2015).

Its dynamics are governed by the transfer operator

$\mathbb{Q}_p$36

For each $\mathbb{Q}_p$37, there exists an absolutely continuous probability measure $\mathbb{Q}_p$38 on $\mathbb{Q}_p$39 with density $\mathbb{Q}_p$40 of bounded variation such that

$\mathbb{Q}_p$41

for all Borel sets $\mathbb{Q}_p$42. The fixed density satisfies

$\mathbb{Q}_p$43

and is bounded away from $\mathbb{Q}_p$44 and from $\mathbb{Q}_p$45 on $\mathbb{Q}_p$46; hence $\mathbb{Q}_p$47 is equivalent to Lebesgue measure. The skew product $\mathbb{Q}_p$48 is mixing with exponential decay of correlations, and it satisfies a central limit theorem and a large deviation principle for observables of bounded variation (Kalle et al., 2015).

The number-theoretic consequences differ sharply from those of regular real continued fractions. For $\mathbb{Q}_p$49-almost every $\mathbb{Q}_p$50, the geometric mean of digits is finite and $\mathbb{Q}_p$51, while the arithmetic mean diverges to $\mathbb{Q}_p$52. If $\mathbb{Q}_p$53 omits at most isolated digits, every $\mathbb{Q}_p$54 admits a semi-regular expansion with all digits in $\mathbb{Q}_p$55; in particular, every $\mathbb{Q}_p$56 has an expansion with only even digits or only odd digits. The set $\mathbb{Q}_p$57 of points admitting an expansion with digits only in $\mathbb{Q}_p$58 has positive Lebesgue measure and is a countable union of intervals, even containing $\mathbb{Q}_p$59 (Kalle et al., 2015). This marks the real random system as a flexible semi-regular analogue of Schneider’s framework rather than the standard $\mathbb{Q}_p$60-adic Schneider map itself.