It is known that winding numbers can be used to prove the existence of fixed points, see the following question Winding Numbers and Fixed Point Theorems. Every fixed point can have a different index, thus a winding number of two might either correspond to two fixed points each with an index of one or to a single fixed point with an index of two. Does anyone know whether an area preserving map (which is also a homeomorphism) can ever have a fixed point with the absolute value of the index greater than 1? Thanks.
2026-04-03 19:09:18.1775243358
Winding Number and Area Preserving Maps
207 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 DYNAMICAL-SYSTEMS
- Stability of system of parameters $\kappa, \lambda$ when there is a zero eigenvalue
- Stability of stationary point $O(0,0)$ when eigenvalues are zero
- Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.
- Question on designing a state observer for discrete time system
- How to analyze a dynamical system when $t\to\infty?$
- The system $x' = h(y), \space y' = ay + g(x)$ has no periodic solutions
- Existence of unique limit cycle for $r'=r(μ-r^2), \space θ' = ρ(r^2)$
- Including a time delay term for a differential equation
- Doubts in proof of topologically transitive + dense periodic points = Devaney Chaotic
- Condition for symmetric part of $A$ for $\|x(t)\|$ monotonically decreasing ($\dot{x} = Ax(t)$)
Related Questions in FIXED-POINT-THEOREMS
- Newton's method with no real roots
- Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.
- Banach and Caristi fixed point theorems
- Show that $\Phi$ is a contraction with a maximum norm.
- Using Fixed point iteration to find sum of a Serias
- Map a closed function $f: (1,4) \rightarrow (1,4)$ without fixed point
- Stop criterium for fixed point methods
- Approximate solutions to nonlinear differential equations using an integral sequence
- Inverse function theorem via degree theory
- Fixed point of a map $\mathbb R^n \rightarrow \mathbb R^n$
Related Questions in WINDING-NUMBER
- how to prove this homotopic problem
- Residue Theorem: Inside vs. Outside
- Evaluating a complex contour integral
- Proof about sum of indices of singular points
- Identifying the winding number of $C$ around points in the regions shown
- Consider homotopy of closed curves. Show equality of winding numbers.
- Hanging picture on the wall with two nails
- Winding number of two closed courves in C with the constraint ∀t:|c1(t)−c2(t)|≠|c1(t)|+|c2(t)|.
- Existence of complex branch for real exponents
- Defining the winding number on a sphere
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?
Yes; it is possible to have fixed points of an arbitrary negative index. To get an example, think first of the index $-1$ fixed point at the origin for the map $(x,y) \mapsto (2x, y/2)$. This fixed point is hyperbolic and its local dynamics consists of four "hyperbolic sectors". You can make examples with more than 4 hyperbolic sectors, to get fixed points of other negative indices. For instance the fixed point in this picture has index $-2$ (and can be made area-preserving). In general if you have a similar dynamics with $2k$ sectors the index will be $1-k$.
So the absolute value of the index may be anything. What is true however is that the index itself (for an area-preserving homeomorphism) can't be greater than $1$. This is proved here:
Pelikan, S., Slaminka, E. (1987). A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds. Ergodic Theory and Dynamical Systems, 7(3), 463-479. https://doi.org/10.1017/S0143385700004132
In the case of $C^1$ diffeomorphisms there is an earlier version (with much easier proof) here:
Simon, Carl P. "A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics.." Inventiones mathematicae 26 (1974): 187-200.