Brjuno Set and Its Dynamical Role

• Brjuno set is the collection of irrational numbers whose continued-fraction denominators grow slowly enough for the defining Brjuno series to converge, marking a sharp arithmetic threshold.
• Its equivalent formulations via the Gauss map and functional equations bridge number theory, ergodic theory, and potential theory to characterize local regularity.
• In holomorphic dynamics, Brjuno numbers determine the linearizability of germs at irrationally indifferent fixed points, influencing the structure of Siegel disks and Julia sets.

The Brjuno set is the arithmetic set of irrational numbers whose continued-fraction denominators grow slowly enough for the Brjuno series to converge. In its classical one-dimensional form,

$$\mathcal B=\left\{\alpha\in \mathbb R\setminus\mathbb Q:\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<\infty\right\},$$

where $P_n/Q_n$ are the convergents of the continued fraction of $\alpha$. It is a central object in small-divisor theory because it gives the sharp arithmetic threshold for local holomorphic linearization at irrationally indifferent fixed points, and it also admits analytic, ergodic, and potential-theoretic characterizations that make the complement $\mathbb R\setminus\mathcal B$ appear extremely thin (Akramov et al., 21 Jul 2025, Balazard et al., 2012).

1. Arithmetic definition and equivalent formulations

For $\alpha\in\mathbb R\setminus\mathbb Q$, write

$$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$

The classical Brjuno condition is

$$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$

and the Brjuno set $\mathcal B$ is the set of irrationals satisfying this condition. Its complement consists of those irrationals for which the same series diverges to $+\infty$ (Akramov et al., 21 Jul 2025).

A function-theoretic formulation uses the Gauss map on

$$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$

with iterates $P_n/Q_n$0, $P_n/Q_n$1. The Brjuno function in the form used by Balazard and Martin is

$P_n/Q_n$2

The irrationals $P_n/Q_n$3 for which $P_n/Q_n$4 are Brjuno numbers, while those with $P_n/Q_n$5 are called Cremer numbers. Proposition 1 in that work yields the standard equivalence

$P_n/Q_n$6

so the continued-fraction and Brjuno-function definitions are identical (Balazard et al., 2012).

The same paper records the exact functional equation

$P_n/Q_n$7

which makes explicit that $P_n/Q_n$8 is the finiteness locus of a recursively defined small-divisor series (Balazard et al., 2012).

2. Position in holomorphic dynamics

The classical dynamical problem is linearization of a germ

$P_n/Q_n$9

near a fixed point with multiplier $\alpha$0. The difficult case is

$\alpha$1

where small divisors appear. In the form recalled in recent work, the Brjuno–Yoccoz theorem states that if $\alpha$2 is Brjuno then the germ is linearizable, while if $\alpha$3 is not Brjuno then the quadratic germ

$\alpha$4

is not linearizable and every neighborhood of the origin contains infinitely many periodic orbits (Akramov et al., 21 Jul 2025).

For the quadratic family

$\alpha$5

the arithmetic meaning is especially sharp: $\alpha$6 is Brjuno if and only if $\alpha$7 is linearizable at $\alpha$8, and the maximal linearization domain is the Siegel disk $\alpha$9. In the high-type subclass

$\mathbb R\setminus\mathcal B$0

the Brjuno case still exhibits thin postcritical geometry: for every Brjuno $\mathbb R\setminus\mathcal B$1, the postcritical set $\mathbb R\setminus\mathcal B$2 has zero area, and for almost every point in the Julia set the $\mathbb R\setminus\mathcal B$3-limit set is exactly $\mathbb R\setminus\mathcal B$4 (Cheraghi, 2012).

The same arithmetic locus persists in more global polynomial settings. For Julia-saturated polynomials, the admissible irrational rotation numbers of Siegel disks are precisely Brjuno rotation numbers; equivalently, such polynomials do not admit exotic Siegel disks with non-Brjuno rotation number. This includes, in particular, the families $\mathbb R\setminus\mathcal B$5 and $\mathbb R\setminus\mathcal B$6 (Geyer, 2015).

3. Real-variable, functional, and local-regularity characterizations

A precise real-variable characterization identifies the Brjuno set with the Lebesgue set of the Brjuno function. Theorem 1 of Balazard and Martin states that the Lebesgue points of $\mathbb R\setminus\mathcal B$7 are exactly the Brjuno numbers: $\mathbb R\setminus\mathcal B$8 Equivalently, Brjuno numbers are precisely the points at which the local symmetric averages of $\mathbb R\setminus\mathcal B$9 tend to $\alpha\in\mathbb R\setminus\mathbb Q$0. In the same framework, if

$\alpha\in\mathbb R\setminus\mathbb Q$1

then $\alpha\in\mathbb R\setminus\mathbb Q$2 is differentiable exactly at Brjuno numbers, while for a Cremer number $\alpha\in\mathbb R\setminus\mathbb Q$3,

$\alpha\in\mathbb R\setminus\mathbb Q$4

The paper also records that the continuity set of $\alpha\in\mathbb R\setminus\mathbb Q$5 is empty, whereas the Lebesgue set, the $\alpha\in\mathbb R\setminus\mathbb Q$6-Lebesgue set, and the differentiability set of $\alpha\in\mathbb R\setminus\mathbb Q$7 all coincide with the Brjuno set (Balazard et al., 2012).

The local singularity structure is explicit near rationals. If $\alpha\in\mathbb R\setminus\mathbb Q$8 with $\alpha\in\mathbb R\setminus\mathbb Q$9, then

$$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$0

so rationals are logarithmic singular centers rather than Lebesgue points. Globally,

$$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$1

for the modulus of continuity of $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$2 (Balazard et al., 2012).

Pointwise regularity of the Brjuno function itself is governed by Diophantine approximation. If $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$3 denotes the Calderón–Zygmund $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$4-exponent of the Brjuno function $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$5, and $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$6 is the irrationality exponent of $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$7, then

$$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$8

In particular, non-Brjuno irrationals have vanishing $$\alpha=[a_0,a_1,a_2,\dots],\qquad \frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n].$$9-exponent and are not Lebesgue points, whereas badly approximable numbers are the points of maximal regularity. The same paper derives the multifractal spectrum

$$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$0

for the Brjuno function (Jaffard et al., 2015).

An extremal fact inside the set is that the Brjuno function attains a strict global minimum at the golden section

$$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$1

For every Brjuno number $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$2, one has $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$3, so $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$4 is the unique global minimizer on the Brjuno set (Balazard et al., 2020).

4. Fine size of the complement

Recent work shows that $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$5 is much smaller than a null set in the ordinary measure-theoretic sense. With respect to the kernel

$$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$6

the complement of the Brjuno set has zero capacity: $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$7 Equivalently, every compact $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$8 satisfies $$\sum_{n=1}^{\infty}\frac{\log Q_{n+1}}{Q_n}<+\infty,$$9. In the potential-theoretic formulation used there, this means that there exists a finite Borel measure whose $\mathcal B$0-potential is $\mathcal B$1 at every point of the non-Brjuno set (Akramov et al., 21 Jul 2025).

The same theorem yields a Hausdorff-measure corollary. For the gauge

$\mathcal B$2

one has

$\mathcal B$3

Since these gauges decay much more slowly than any power $\mathcal B$4, the vanishing of $\mathcal B$5 is substantially stronger than zero Lebesgue measure or a small Hausdorff-dimension statement. The proof is arithmetic and constructive: it converts divergence of the Brjuno sum into divergence of a stronger weighted series, then constructs a finite atomic measure on the rationals whose potential diverges at every non-Brjuno point (Akramov et al., 21 Jul 2025).

This capacity-zero theorem is presented as an improvement of the earlier Sadullaev–Rakhimov result on the capacity dimension of the Brjuno set, and it leaves open whether the threshold $\mathcal B$6 is optimal (Akramov et al., 21 Jul 2025).

5. Alternative continued fractions and robustness of the definition

The Brjuno set is robust under several changes of continued-fraction algorithm. For Nakada’s $\mathcal B$7-continued fractions with $\mathcal B$8, and for weights $\mathcal B$9 satisfying

$+\infty$0

the generalized Brjuno functions $+\infty$1 differ from $+\infty$2 by a bounded amount. Consequently, the set of $+\infty$3-Brjuno numbers does not depend on $+\infty$4 as long as $+\infty$5. The by-excess case $+\infty$6 is exceptional: for the logarithmic weight, a real number is a classical Brjuno number if and only if both $+\infty$7 and $+\infty$8 are semi-Brjuno for the by-excess algorithm (0705.1690).

More refined comparisons are available for the by-excess, odd, even, and odd-odd continued fractions. For the odd continued fraction, the odd Brjuno function differs from the classical Brjuno function by a Hölder-continuous correction, so the odd Brjuno set coincides with the classical Brjuno set. For the by-excess algorithm, the correct comparison involves the even part $+\infty$9, whose finiteness is equivalent to classical Brjuno membership. The even algorithm behaves differently: the paper proves that

$$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$0

is bounded, so the classical Brjuno condition is equivalent to the simultaneous finiteness of the even Brjuno function and the odd-odd Brjuno-type function, rather than to finiteness of the even Brjuno function alone (Lee et al., 2021).

A complementary analytic decomposition comes from the semi-Brjuno function $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$1, defined via the by-excess map. The Brjuno and Wilton functions are expressed, up to bounded defects, through the even and odd parts of $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$2: $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$3 For the Brjuno side, the defect

$$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$4

is Hölder continuous of exponent $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$5, which isolates the main arithmetic singularity in the even semi-Brjuno core rather than in the correction term (Burrin et al., 11 Mar 2025).

6. Higher-dimensional, simultaneous, and computational extensions

Several papers extend the Brjuno-set paradigm beyond one real parameter. In $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$6, one defines

$$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$7

and the multidimensional Brjuno set

$$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$8

This is genuinely several-variable: coordinatewise Brjuno behavior does not suffice. The complement $$X=(0,1)\setminus\mathbb Q,\qquad \alpha(x)=\{1/x\},$$9 has zero capacity with respect to

$P_n/Q_n$00

and therefore zero Hausdorff measure for the gauges

$P_n/Q_n$01

This generalizes the one-dimensional capacity theorem to higher dimensions (Akramov et al., 24 May 2025).

For families of commuting germs, the arithmetic object becomes a simultaneous Brjuno-type set of eigenvalue data. If $P_n/Q_n$02 are the spectra of the linear parts, the relevant divisor is

$P_n/Q_n$03

and the simultaneous Brjuno condition requires convergence of

$P_n/Q_n$04

where $P_n/Q_n$05 is the minimum of $P_n/Q_n$06 over non-simultaneously resonant multi-indices $P_n/Q_n$07 with $P_n/Q_n$08. Under simultaneous diagonalizability, this condition is sufficient for simultaneous holomorphic linearization; together with commutativity, it yields an if-and-only-if criterion for simultaneous holomorphic linearizability (Raissy, 2010).

The label “Brjuno” also extends to Gevrey regularity. For $P_n/Q_n$09, the class $P_n/Q_n$10 of $P_n/Q_n$11-Brjuno vectors in $P_n/Q_n$12 admits equivalent formulations through small divisors, best-approximation vectors, and renormalization stopping times, and it is the arithmetic condition used to linearize $P_n/Q_n$13-Gevrey torus flows sufficiently close to constant vector fields. The case $P_n/Q_n$14 recovers the classical Brjuno condition in the sense stated in that paper (Dias et al., 2017).

Finally, recent computability work shows that the values of Brjuno-like functions on computable inputs can still encode noncomputable information. For the classical Yoccoz-Brjuno function, if

$P_n/Q_n$15

then every left-computable $P_n/Q_n$16 is realized as $P_n/Q_n$17 for some computable $P_n/Q_n$18. More generally, there exist points at which Brjuno-like functions are not computable at all. That work concerns value-complexity rather than a complete effective classification of the Brjuno set as a subset of $P_n/Q_n$19 (Shevchenko et al., 8 Jan 2025).