Perez-Marco Set in Holomorphic Dynamics

• Perez-Marco Set is defined as a collection of irrational rotation numbers satisfying a specific continued-fraction summability condition that forces infinitely many periodic orbits in non-linearizable germs.
• It unifies arithmetic analysis and geometric constructions by linking refined Diophantine conditions with invariant continua (hedgehogs) in the dynamics near an indifferent fixed point.
• Potential theory shows that the complement of the Perez-Marco set has zero capacity, emphasizing its fine threshold between linearizable and non-linearizable dynamics.

The Pérez–Marco set arises in the local dynamics of holomorphic germs with an irrationally indifferent fixed point and appears in two closely related senses in the literature. In the arithmetic sense, it is the set $\mathcal{PM}\subset \mathbb{R}\setminus\mathbb{Q}$ of rotation numbers satisfying a continued-fraction summability condition that governs the ubiquity of small periodic orbits for non-linearizable germs (Akramov et al., 21 Jul 2025). In the geometric sense, dynamical papers also use “Perez-Marco set of $f$” for the minimal full invariant continuum, or hedgehog, attached to a specific non-linearizable germ (Biswas, 2010). These usages are linked by Pérez-Marco’s analysis of irrationally indifferent dynamics: the arithmetic condition selects those angles for which every non-linearizable germ necessarily exhibits infinitely many periodic orbits accumulating at the fixed point, while the geometric construction describes the invariant continua that organize such dynamics (Akramov et al., 21 Jul 2025).

1. Arithmetic definition via continued fractions

Let $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and write its principal continued-fraction convergents by

$\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$

where

$$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$

The arithmetic Pérez–Marco condition is

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

and the corresponding set is

$$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$

This is the “so-called Pérez-Marco set” in the sense of rotation angles (Akramov et al., 21 Jul 2025).

Within the same source, this condition is described as “a weaker arithmetic condition than the classical Brjuno condition.” The relation to Brjuno numbers is explicit: every Brjuno number is Pérez-Marco, and this inclusion is used to derive a corresponding smallness statement for the complement of the Brjuno set (Akramov et al., 21 Jul 2025). This suggests that $\mathcal{PM}$ is designed to isolate a finer threshold for a specific dynamical phenomenon than the one captured by Brjuno’s linearization criterion.

2. Dynamical characterization for irrationally indifferent germs

The relevant local model is a holomorphic germ near a fixed point at the origin,

$f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$

with $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ (Akramov et al., 21 Jul 2025). A central problem is to determine for which rotation angles $f$0 every non-linearizable germ with multiplier $f$1 must nevertheless exhibit infinitely many small periodic orbits accumulating at $f$2.

Pérez-Marco’s theorem, as summarized in the capacity paper, gives an exact arithmetic characterization. If $f$3, then every non-linearizable germ

$f$4

has infinitely many periodic orbits accumulating at $f$5. Conversely, if $f$6, then one can construct a non-linearizable germ with no small periodic points beyond the fixed point (Akramov et al., 21 Jul 2025). In this formulation, $f$7 is not merely a Diophantine class; it is the parameter set for a universal dynamical forcing statement.

This characterization locates the Pérez–Marco condition between two standard themes in one-dimensional holomorphic dynamics: linearization at irrationally indifferent fixed points and the structure of non-linearizable, or Cremer-type, dynamics. The condition does not assert linearizability. Instead, it controls what must occur when linearization fails.

3. Capacity-theoretic smallness of the complement

Akramov–Ashirov study the size of the exceptional set $f$8 through a fine potential-theoretic capacity associated with the kernel

$f$9

For a compact set $\alpha\in\mathbb{R}\setminus\mathbb{Q}$0 and a probability measure $\alpha\in\mathbb{R}\setminus\mathbb{Q}$1 supported on $\alpha\in\mathbb{R}\setminus\mathbb{Q}$2, they define the $\alpha\in\mathbb{R}\setminus\mathbb{Q}$3-potential

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

the mutual energy

$\alpha\in\mathbb{R}\setminus\mathbb{Q}$5

the extremal quantity

$\alpha\in\mathbb{R}\setminus\mathbb{Q}$6

and the capacity

$\alpha\in\mathbb{R}\setminus\mathbb{Q}$7

This extends to an outer capacity on Borel sets $\alpha\in\mathbb{R}\setminus\mathbb{Q}$8 by

$\alpha\in\mathbb{R}\setminus\mathbb{Q}$9

A fundamental criterion is that $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$0 if and only if there exists a finite Borel measure $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$1 with $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$2 for all $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$3 (Akramov et al., 21 Jul 2025).

The main theorem is

$\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$4

In particular,

$\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$5

where the gauge function is

$\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$6

The paper summarizes this by saying that the complement is “extremely thin—thinner than any power-law or even any fractional-log gauge” (Akramov et al., 21 Jul 2025).

4. Proof architecture of the zero-capacity theorem

The proof is organized as a divergence argument coupled to an explicit measure construction (Akramov et al., 21 Jul 2025). The starting point is the observation that if $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$7, then

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

A delicate elementary lemma then yields a stronger divergence statement for a weighted series involving $\frac{P_n}{Q_n}=[a_0,a_1,\dots,a_n],$9, $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$0, $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$1, and $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$2, for arbitrary $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$3.

The potential-theoretic step is the construction of a discrete “Dirac-comb” measure supported on rational points $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$4, with weights of order $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$5. The finiteness of the measure follows from the convergence of $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$6 (Akramov et al., 21 Jul 2025).

To estimate the potential at $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$7, one uses the convergents $$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$8 and the Diophantine estimate

$$A(\alpha_n)=\frac{1}{\alpha_n}-\left\lfloor \frac{1}{\alpha_n}\right\rfloor,\qquad \alpha_0=\alpha-\lfloor\alpha\rfloor,\qquad \alpha_{n+1}=A(\alpha_n),\qquad a_{n+1}=\left\lfloor \frac{1}{\alpha_n}\right\rfloor.$$9

Each corresponding contribution to $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$0 is then bounded below by the weighted terms furnished by the divergence lemma, so the full series diverges and

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

for every $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$2. By the capacity criterion, this implies

$$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$3

for all $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$4 (Akramov et al., 21 Jul 2025).

The significance of the argument is methodological as well as quantitative. It converts a continued-fraction divergence condition into a potential-theoretic extinction statement for the exceptional set.

5. Hedgehogs and the “Perez-Marco set of $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$5”

In the geometric literature on irrationally indifferent germs, the term “Perez-Marco set” also refers to an invariant compact associated with a fixed germ. For

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

one asks whether $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$7 is analytically conjugate to the rigid rotation

$$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$8

If so, $$\sum_{n=1}^\infty \frac{\ln\ln Q_{n+1}}{Q_n}<+\infty,$$9 is linearizable and its maximal domain of conjugacy is the Siegel disk. A compact set $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$0 is called a Siegel compactum for $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$1 if it is full, connected, invariant, contains $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$2, and contains a neighborhood of $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$3 whenever $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$4 is linearizable (Biswas, 2010).

Perez-Marco’s uniqueness theorem states that for an irrationally indifferent germ there exists a unique minimal full invariant continuum $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$5 containing $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$6, and any full continuum $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$7 with $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$8 and $$\mathcal{PM}=\left\{\alpha\in\mathbb{R}\setminus\mathbb{Q}:\sum_n\frac{\ln\ln Q_{n+1}}{Q_n}<\infty\right\}.$$9 must contain $\mathcal{PM}$0 (Biswas, 2010). In the non-linearizable case, this nontrivial compact of empty interior is called the hedgehog or Perez-Marco set of $\mathcal{PM}$1.

This germ-specific object should be distinguished from the arithmetic set $\mathcal{PM}$2. The former is a continuum in the dynamical plane of a particular germ; the latter is a subset of irrational rotation numbers. The relationship is conceptual rather than notational: the arithmetic condition identifies those angles for which non-linearizable germs must have recurrent small-cycle structure, while the hedgehog construction describes the invariant continua that occur in non-linearizable dynamics.

The geometric theory is flexible. There exists a non-linearizable germ whose common hedgehog has strictly positive Lebesgue area, and there exists a non-linearizable germ with a hedgehog for which the fixed point $\mathcal{PM}$3 is inaccessible from $\mathcal{PM}$4 (Biswas, 2010). These results indicate that hedgehogs can have highly nontrivial measure-theoretic and topological properties.

6. Consequences, open problems, and bibliographic clarification

The zero-capacity theorem has several explicit consequences. The complement $\mathcal{PM}$5 is “polar of higher order” and, in particular, pluripolar in $\mathcal{PM}$6. By comparison between the fine capacities and Hausdorff measures, one also obtains

$\mathcal{PM}$7

for the gauge $\mathcal{PM}$8 displayed above (Akramov et al., 21 Jul 2025). In the language of that paper, $\mathcal{PM}$9 is “thick” enough to carry all such size, while its complement is negligible in these potential-theoretic senses.

Several open problems are recorded. One is to determine, for each $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$0, the exact critical exponent $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$1 such that $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$2 jumps from positive to zero as $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$3 passes above $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$4; the present proof gives $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$5. Further questions ask whether the capacity-zero complement also carries zero harmonic measure for natural parameter slices, whether it is dynamically negligible in other senses such as prevalence under random perturbations, and how to extend these capacity results to multi-dimensional small-divisor problems, where continued-fraction methods do not directly generalize (Akramov et al., 21 Jul 2025).

A terminological clarification is necessary. In arithmetic papers, “Pérez-Marco set” denotes the subset $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$6 of irrational angles. In dynamical papers such as the hedgehog construction, “Perez-Marco set of $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$7” denotes the minimal full invariant continuum attached to a specific non-linearizable germ (Biswas, 2010). The two usages are related but not identical.

A bibliographic clarification is also warranted. The record for “Capacity of the $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$8-Brjuno-Rüssmann set” (Akramov, 18 Oct 2025) contains an abstract stating a theorem about the complement of the Perez-Marco set, but the manuscript details supplied for that record state that the text “does not discuss the Perez–Marco set at all” and that its definitions and proofs concern Brjuno and $f(z)=e^{2\pi i\alpha}z+c_2z^2+c_3z^3+\cdots,$9-Brjuno-Rüssmann sets instead (Akramov, 18 Oct 2025). For the Pérez–Marco set itself, the substantive capacity statement in the materials considered here is the theorem of Akramov–Ashirov in “On the capacity dimensions of the Brjuno and Perez-Marco sets” (Akramov et al., 21 Jul 2025).