0
votes
1answer
27 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
0
votes
0answers
36 views

Parabolic PDEs and Gradient Systems

Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite ...
0
votes
1answer
108 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
1
vote
0answers
47 views

Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
2
votes
1answer
63 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
2
votes
1answer
115 views

Properties of this Sturm-Liouville problem.

Given the ODE $$f''(x) + \left(\alpha_1 \cos(x) + \cos^2(x) - \lambda \right) f(x)= 0,$$ where $\theta \in [-\pi,\pi]$, $||f||_{L^2} < \infty$. I was wondering whether there is anything we ...
1
vote
0answers
26 views

Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
0
votes
0answers
56 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
3
votes
0answers
27 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
4
votes
0answers
60 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
1
vote
1answer
46 views

compactness of the set of invariant measures

Suppose that we have a dynamical system on some compact space $X$ with discrete time space and transformation given by some $\phi : X \rightarrow X$. My question is when is the set $Prob(\phi)$ of all ...
1
vote
0answers
31 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
1
vote
1answer
42 views

A set of functions which is open in the space $C^1[0,1]$

Let $f:[0,1]\to [0,1]$ be a $C^1$ and increasing function such that $i)$ If $f(p)=p$ then $|f'(p)|\ne 1$ I want to prove that there exist an $\varepsilon>0$ such that if $g\in C^1$ and ...
4
votes
0answers
88 views

An element of $\ell^2$ wanted

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is ...
2
votes
1answer
73 views

Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic

Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
3
votes
1answer
56 views

Why should we use inverse homemorphism here?

I am a brand-new comer in dynamical system. I find it interesting that when defining ergodicity of classical dynamical system $(X,\sigma)$, they use $\mu(\sigma^{-1}(E))$ there. Since $\sigma$ is a ...
0
votes
0answers
65 views

Ultrafilters and Birkhoff averages

Dear Andreas Blass: The example you presented may be seen as a sequence $[f(T^n(x))]_n$, where $f: X \to R$ and $T: X \to X$ are measurable (are even better than that). For those $f$ and $T$, the ...
2
votes
0answers
75 views

Decreasing function with a fixed point and 2 cycle?

Can you give me an example of a decreasing function with a fixed point and 2-cycle?
2
votes
1answer
58 views

An abstract $\alpha$-contracting dynamical system

$\newcommand{\f}{\phi}$$\newcommand{\ep}{\varepsilon}$$\newcommand{\R}{\mathbb R}$ Suppose $(\f_t)_{t\ge0}$ is an abstract dynamical system in a Banach space $(X,\|\mathord\cdot\|)$. Let $C(x,\ep)$ ...
2
votes
1answer
70 views

Perturbing an abstract discrete dynamical system

Let $X$ be a Banach space. Denote for $x_0\in X$ and $r>0$ the closed ball centered at $x_0$ by $B(x_0,r)=\lbrace x\in X:\|x-x_0\|\le r\rbrace$. Suppose $f:X\to X$ a bounded map with a fixed point ...
1
vote
0answers
191 views

Criterion for a countable family of functions to separate points

So this may be a weird/dumb question, but I was wondering whether a sufficient condition is known for a (countable) family of continuous functions to separate points on, say, a compact metric space ...
3
votes
0answers
235 views

Definitions of weak (topological) mixing

Let $X$ be a compact (metric) space and $T:X\rightarrow X$ be a continuous map. Let $U_T:C(X)\rightarrow C(X)$ be the linear operator $U_T(f) = f\circ T$. Then Wikipedia (see ...