Intereting Posts

$m\{x\in :f'(x)=0\}>0$
Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.
Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$
Prove that the limit of $\sqrt{n+1}-\sqrt{n}$ is zero
How to show that if all fourier coefficient of a function is zero, then the function is zero function?
Summing the cubes of the insertion sequence
Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves
Differential forms turn infinitesimal stuff rigorous?
What's the significance of Tate's thesis?
Isomorphism of an endomorphism ring, how can $R\cong R^2$?
Showing that group of orientation preserving isometries of Icosahedron is a simple group
Reasoning that $ \sin2x=2 \sin x \cos x$
Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down
Proving Nonhomogeneous ODE is Bounded

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$

My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with length 2 exists.

May you could help me to prove this statement?

- Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs
- Example of dynamical system where: $NW(f) \not\subset \overline{R(f)}$
- How can a Markov chain be written as a measure-preserving dynamic system
- An issue with approximations of a recurrence sequence
- Non wandering Set
- Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

- About Collatz 3n+3?
- The fractional parts of the powers of the golden ratio are not equidistributed in
- does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?
- Non wandering Set
- Compact space, continuous dynamical system, stationary point
- To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$
- System of equations, limit points
- Omega limit set is invariant
- Are the fractional parts of $\log \log n!$ equidistributed or dense in $$?
- Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

In general if $f$ is continuous, and there exists an integer k>1 and a real number $a$ such that $f^k(a)=a$ then there exists a real number $r$ such that $f(r)=r$.

**Proof:** Let $g(x)=f(x)-x$. Since:

$g(a)+g(f(a))+g(f^2(a))+…+g(f^{k-1}(a))=f^k(a)-a=0$, therefore either one of the numbers $g(a),g(f(a)),g(f^2(a)),…,g(f^{k-1}(a))$ is zero (in this case we are done) or one of these numbers is positive and another number is negative. Assume WLOG that the second case holds. Let $g(f^i(a))g(f^j(a))<0$ (it means that they have different signs). Since, $g$ has a sign change in the interval $[\min(f^i(a),f^j(a)),\max(f^i(a),f^j(a))]$, thus $g$ has a root.

Also, if you like hitting flies with sledgehammers, this directly follows from Sharkovskii’s theorem.

- Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
- solve $x^2 – 25 xy + y^2 = 1$ does it have a solution?
- How do proof verifiers work?
- On an expansion of $(1+a+a^2+\cdots+a^n)^2$
- Other interesting consequences of $d=163$?
- Let $f: V_n \to V_n$ be an endomorphism, prove $\text{dim}(\text{Ker}f \cap \text{Im} f) = r(f) – r(f^2)$
- Is there any “superlogarithm” or something to solve $x^x$?
- Integral of $\frac{\sqrt{x^2+1}}{x}$
- Fourier Transform of $f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$
- Convergence in law and uniformly integrability
- An integral for the New Year
- Good books on Philosophy of Mathematics
- $R/M$ is a division ring
- Are half of all numbers odd?
- Is trace of regular representation in Lie group a delta function?