Real Non-Attractive Fixed Point Conjecture

• The Real Non-Attractive Fixed Point Conjecture is a fixed point property in holomorphic dynamics stating that every polynomial of degree ≥2 has a fixed point with a multiplier whose real part is at least 1.
• The proof utilizes the residue fixed point index and holomorphic fixed point formula, establishing a global identity that ensures at least one fixed point meets the multiplier inequality.
• Extensions to rational maps (with a super-attracting fixed point) and to harmonic functions via induced h-fixed points highlight parallel techniques and deepen understanding of fixed point multiplier geometry.

Searching arXiv for papers on the Real Non-Attractive Fixed Point Conjecture and closely related fixed-point literature. The Real Non-Attractive Fixed Point Conjecture is a problem in holomorphic dynamics concerning the distribution of multipliers of fixed points of complex polynomials. In its classical form, attributed to Coelho and Kalantari, it asserts that every polynomial $P$ of degree at least $2$ has a fixed point $z_0$ such that $\operatorname{Re}(P'(z_0))\ge 1$. The conjecture has been settled affirmatively for polynomials and, more generally, for rational maps with a super-attracting fixed point (Kumar et al., 2020). More recently, a harmonic analogue has been formulated and proved for complex harmonic polynomial and rational functions by replacing ordinary fixed points with $\mathfrak h$-fixed points adapted to the decomposition $f=h+\overline g$ (Vaseem, 24 Jul 2025).

1. Classical formulation and terminology

The conjecture concerns fixed points of a holomorphic self-map. For a rational map $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$, a fixed point is a point $z_0$ with $R(z_0)=z_0$, and its multiplier is $\lambda=R'(z_0)$ for finite fixed points. In the polynomial case, “fixed point” ordinarily means a finite fixed point unless explicitly stated otherwise (Kumar et al., 2020).

In this setting, the conjecture is:

$2$0

The inequality is weak. This is essential, because equality occurs at multiple fixed points: a fixed point is multiple if and only if its multiplier is $2$1. Accordingly, the conjecture cannot be sharpened in general to $2$2 (Kumar et al., 2020).

The adjective “real” does not refer to real-variable dynamics or to maps $2$3. It refers to the real part of the multiplier. The phrase “non-attractive” is likewise specialized: the conjecture does not ask for a repelling fixed point in the usual modulus sense, but for a fixed point whose multiplier lies in the half-plane $2$4. The paper also recalls the standard notion of weakly repelling fixed point, meaning $2$5 or $2$6, but the conjecture itself is formulated in terms of $2$7 rather than $2$8 (Kumar et al., 2020).

2. Resolution for polynomials and extension to rational maps

The decisive result is stronger than the original polynomial statement. Every rational map of degree at least two with a super-attracting fixed point has a fixed point whose multiplier has real part at least $2$9; since every polynomial of degree at least $z_0$0 has $z_0$1 as a super-attracting fixed point, the polynomial case follows immediately (Kumar et al., 2020).

The proof is based on the residue fixed point index and the holomorphic fixed point formula. For a fixed point $z_0$2 of a rational map $z_0$3, the residue fixed point index is

$z_0$4

and for a simple fixed point with multiplier $z_0$5,

$z_0$6

The global identity is

$z_0$7

If one fixed point is super-attracting, then its multiplier is $z_0$8, hence its index contributes $z_0$9. For the remaining fixed points with multipliers $\operatorname{Re}(P'(z_0))\ge 1$0, one obtains

$\operatorname{Re}(P'(z_0))\ge 1$1

Taking real parts yields

$\operatorname{Re}(P'(z_0))\ge 1$2

If every $\operatorname{Re}(P'(z_0))\ge 1$3, each summand is strictly positive, which is impossible. Therefore at least one fixed point satisfies $\operatorname{Re}(P'(z_0))\ge 1$4 (Kumar et al., 2020).

This argument also explains the precise scope of the theorem. The super-attracting hypothesis is indispensable in the rational setting. The map

$\operatorname{Re}(P'(z_0))\ge 1$5

has fixed points at the $\operatorname{Re}(P'(z_0))\ge 1$6-th roots of unity, and each fixed point has multiplier $\operatorname{Re}(P'(z_0))\ge 1$7, which is real and $\operatorname{Re}(P'(z_0))\ge 1$8. Thus the conjectural conclusion fails for general rational maps without a super-attracting fixed point (Vaseem, 24 Jul 2025).

3. Multiplier geometry and extremal configurations

After settling the conjecture, the analysis in the polynomial case turns to the geometry of fixed-point multipliers. One immediate consequence of the residue identity is that not all fixed-point multipliers of a polynomial can have real part $\operatorname{Re}(P'(z_0))\ge 1$9; if some do, then another must have real part $\mathfrak h$0. The same computation also yields

$\mathfrak h$1

so if all fixed points are simple, then there is a fixed point whose multiplier has nonnegative imaginary part (Kumar et al., 2020).

Several extremal families are characterized explicitly. For quadratic polynomials in normalized form $\mathfrak h$2, the fixed points satisfy

$\mathfrak h$3

and the corresponding multipliers are

$\mathfrak h$4

From this, one obtains the exact criterion: all fixed-point multipliers have real part $\mathfrak h$5 if and only if $\mathfrak h$6. At $\mathfrak h$7, the two fixed points coalesce into a multiple fixed point with multiplier $\mathfrak h$8 (Kumar et al., 2020).

For cubic polynomials with three distinct fixed points, all multipliers have real part $\mathfrak h$9 if and only if the fixed points are collinear and the real part of one multiplier is $f=h+\overline g$0. Via affine normalization, this reduces to a cubic with fixed points $f=h+\overline g$1 and multipliers determined by a parameter $f=h+\overline g$2, with the condition becoming $f=h+\overline g$3 and $f=h+\overline g$4 purely imaginary (Kumar et al., 2020).

The same paper also studies fixed-point multipliers equidistant from $f=h+\overline g$5. For cubic polynomials with simple fixed points, this is equivalent to the fixed points forming an equilateral triangle. For quartics, the supplied text records a necessary condition: if all simple fixed-point multipliers are equidistant from $f=h+\overline g$6, then the four fixed points are vertices of a rectangle. The supplied discussion also notes a textual subtlety: the quartic statement is printed one-way, even though the surrounding exposition suggests a stronger converse was intended (Kumar et al., 2020).

4. Harmonic analogue and $f=h+\overline g$7-fixed points

A harmonic extension of the conjecture was developed for maps of the form

$f=h+\overline g$8

where $f=h+\overline g$9 and $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$0 are analytic. In this setting, the ordinary fixed-point equation $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$1 is not used as the principal notion. Instead, the relevant concept is the $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$2-fixed point: a complex number $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$3 is a finite $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$4-fixed point if

$R:\widehat{\mathbb C}\to\widehat{\mathbb C}$5

In particular, if $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$6 and $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$7, then $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$8 is an induced $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$9-fixed point (Vaseem, 24 Jul 2025).

At such a point, the paper defines two multipliers,

$z_0$0

The harmonic analogue of the conjecture is then a pair of real-part inequalities. The main theorem states that every rational harmonic function $z_0$1, where both $z_0$2 and $z_0$3 are rational of degree at least $z_0$4 and have a super-attracting fixed point, has a $z_0$5-fixed point $z_0$6 such that

$z_0$7

For polynomial harmonic functions, the same conclusion holds without any additional hypothesis, because each polynomial component has $z_0$8 as a super-attracting fixed point (Vaseem, 24 Jul 2025).

The proof does not construct a genuinely harmonic fixed-point index theory. Instead, it reduces the problem to the two analytic components. One proves separately that $z_0$9 has a fixed point $R(z_0)=z_0$0 with $R(z_0)=z_0$1 and that $R(z_0)=z_0$2 has a fixed point $R(z_0)=z_0$3 with $R(z_0)=z_0$4, using the same residue-based holomorphic argument as in the rational theorem. These are then combined into the induced $R(z_0)=z_0$5-fixed point $R(z_0)=z_0$6 (Vaseem, 24 Jul 2025).

The harmonic paper also analyzes the quadratic family

$R(z_0)=z_0$7

If $R(z_0)=z_0$8, then there is a single $R(z_0)=z_0$9-fixed point and its multipliers are $\lambda=R'(z_0)$0. If $\lambda=R'(z_0)$1, there are two simple $\lambda=R'(z_0)$2-fixed points, and the real parts of their multipliers equal $\lambda=R'(z_0)$3 exactly when $\lambda=R'(z_0)$4 (Vaseem, 24 Jul 2025).

5. Scope, limitations, and open directions

The proven holomorphic theorem is exact in two ways. First, it settles the polynomial case completely. Second, it extends to rational maps precisely under a super-attracting fixed-point hypothesis, and the counterexample $\lambda=R'(z_0)$5 shows that no unconditional rational statement of the same form can hold (Kumar et al., 2020).

Equality cases are intrinsic to the theory. If a polynomial or rational map has a multiple fixed point, then the multiplier at that point is $\lambda=R'(z_0)$6, so the conclusion $\lambda=R'(z_0)$7 may be attained sharply. This explains why the statement is formulated with $\lambda=R'(z_0)$8 rather than $\lambda=R'(z_0)$9 (Kumar et al., 2020).

The harmonic extension has a similarly delimited scope. It covers polynomial harmonic functions and rational harmonic functions with super-attracting component maps, but it does not claim a general theorem for arbitrary rational harmonic maps without such a hypothesis. The paper states this explicitly and raises transcendental harmonic functions as an open problem. As a cautionary example, it notes that for

$2$00

no finite $2$01-fixed points exist, so the polynomial and rational argument cannot be extended mechanically to the transcendental setting (Vaseem, 24 Jul 2025).

The conceptual limitation of the harmonic theorem is equally clear. It is a harmonic analogue through decomposition, not a full dynamical theorem for iteration of general harmonic maps. The proof uses the separate holomorphic dynamics of $2$02 and $2$03, rather than a local harmonic dynamical theory based on the Jacobian

$2$04

That formula is standard in harmonic mapping theory, but the paper does not use it in the proof (Vaseem, 24 Jul 2025).

6. Broader usage of the phrase and distinct fixed-point contexts

Outside holomorphic and harmonic dynamics, the expression “real non-attractive fixed point” can arise in a broader descriptive sense: a fixed point with real coordinates or couplings that is not attractive under the relevant flow. This usage is conceptually related but mathematically distinct from the conjecture about $2$05.

An instructive example comes from functional renormalization-group studies of four-dimensional Einstein–Hilbert gravity. A phase diagram derived from the non-perturbative graviton propagator contains several real fixed points, including a Gaussian fixed point at

$2$06

which the paper explicitly calls a “well-known repulsive Gaussian IR fixed point,” alongside a real ultraviolet non-Gaussian fixed point and real infrared-attractive fixed points (Christiansen et al., 2012). In that RG sense, the work provides a concrete example showing that real fixed points need not be attractive.

A related large-$2$07 tensor-field-theory study reports a line of fixed points in a long-range quartic $2$08 model and several IR-stable real fixed points in other tensor models. A line of fixed points implies marginal directions, so such fixed points are not fully attractive in all coupling directions (Harribey, 2022). This suggests that the phrase can also function as shorthand for stability questions in RG theory rather than for the multiplier inequality of the original conjecture.

The established mathematical content of the Real Non-Attractive Fixed Point Conjecture, however, remains the holomorphic and harmonic story: existence of fixed points whose multipliers have real part at least $2$09, first for polynomials, then for rational maps with a super-attracting fixed point, and finally for $2$10-fixed points of polynomial and rational harmonic maps (Kumar et al., 2020).