The Mandelbrot set is known to be connected but whether it is path-connected is an open question. But what is the general consensus/belief among mathematicians? I am unable to convince myself either way because I am rather new to topology and don't yet have a good intuition about connectedness and path-connectedness.
2026-02-22 21:03:04.1771794184
Is the Mandelbrot set path-connected?
1.5k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in COMPLEX-ANALYSIS
- Minkowski functional of balanced domain with smooth boundary
- limit points at infinity
- conformal mapping and rational function
- orientation of circle in complex plane
- If $u+v = \frac{2 \sin 2x}{e^{2y}+e^{-2y}-2 \cos 2x}$ then find corresponding analytical function $f(z)=u+iv$
- Is there a trigonometric identity that implies the Riemann Hypothesis?
- order of zero of modular form from it's expansion at infinity
- How to get to $\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz =n_0-n_p$ from Cauchy's residue theorem?
- If $g(z)$ is analytic function, and $g(z)=O(|z|)$ and g(z) is never zero then show that g(z) is constant.
- Radius of convergence of Taylor series of a function of real variable
Related Questions in FRACTALS
- does the area converge?
- "Mandelbrot sets" for different polynomials
- Is the Mandelbrot set path-connected?
- Does the boundary of the Mandelbrot set $M$ have empty interior?
- What sort of function is this? (Logistic map?)
- effective degree for normalized escape-time of hybrids
- Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$
- A closed form for the sum $\sum_{s=0}^{n-1} e^{\frac{s(s+1)}{2}i\theta}$?
- Given a real number $d , (1<d<2)$, is there a fractal with fractal dimension $d$?
- How can one write a line element for non-integer dimensions?
Related Questions in COMPLEX-DYNAMICS
- effective degree for normalized escape-time of hybrids
- Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$
- Is this proof of Milnor's lemma valid? the one about Newtons method and super-attractive fixed-points corresponding to simple roots
- Hausdorff Dimension of Julia set of $z^2+2$?
- How to find the set of $c$ for which the Julia set of $x^2+c$ completely lies in $\mathbb{R}$?
- Iteration of mapping with nested iterates in logarithm
- Help understanding a proof on why the Mandelbrot set is fractal
- bounds on dimension of Julia sets inside Mandelbrot set
- How does this algorithm get the limit set of "kissing" Schottky group?
- Julia set property: $J(f) = J(f^p)$
Related Questions in PATH-CONNECTED
- Why the order square is not path-connected
- Prove that $\overline S$ is not path connected, where $S=\{x\times \sin(\frac1x):x\in(0,1]\}$
- Is the Mandelbrot set path-connected?
- Example of a topological space that is connected, not locally connected and not path connected
- Example of path connected metric space whose hyperspace with Vietoris topology is not path connected?
- Proof explanation to see that subset of $\mathbb{R}^2$ is not path connected.
- Connectedness and path connectedness of a finer topology
- Show that for an abelian countable group $G$ there exists a compact path connected subspace $K ⊆ \Bbb R^4$ such that $H_1(K)$ isomorphic to $G$
- How to construct a path between two points in a general $n-surface$?
- Definition of product implies the existence of continuous function?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
I cannot speak about the whole of the complex dynamics community of course.
If you are new to the notions of path-connectedness and local connectedness the Mandelbrot set might not be the best topic to gain intuition, since as you said those are current research areas and therefore well beyond the scope of a first course in topology.
That said, there are results (mainly due to JC. Yoccoz, a Fields medallist) proving that the Mandelbrot set is locally connected at every point except possibly some special points (called infinitely renormalizable parameters; roughly speaking they are those points that are in infinitely many nested small copies of the Mandelbrot set). That seems like a good step towards proving the conjecture.
Note that experts are not so much interested in this question by itself (local connectedness of the Mandelbrot set) but rather are interested in a conjecture called genericity of hyperbolicity, which is known to be true IF the Mandelbrot set is locally connected.
Oh and by the way I don't think there is an easy way to convince yourself one way or another (short of reading several research papers on the subject).
For the sake of completeness, here is an argument against the conjecture: it is known that the analogue of the Mandelbrot set for cubic polynomials (the connectedness locus of the Julia set) is NOT locally connected.