All Questions
Tagged with ds.dynamical-systems fa.functional-analysis
118 questions
1
vote
0
answers
97
views
Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
1
vote
0
answers
64
views
Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
0
votes
0
answers
55
views
Dense stratification of a separable Hilbert space
Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
- 5,405
3
votes
1
answer
146
views
spectrum of multiplicative morphisms
Let $T:[0,1]\to[0,1]$ be a continuous map, which is neither surjective nor injective. Put
$$
C([0,1])\ni f\mapsto \Phi(f):=f\circ T\in C([0,1]).
$$
Notice that, under the above conditions, $0\in\sigma(...
- 41
-1
votes
1
answer
74
views
Invariant ergodic measure Volterra operator
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability ...
- 5,405
3
votes
1
answer
291
views
Invariant measure for composition on space of continuous functions
Let $C_g:C(\mathbb{R}^d;\mathbb{R}^d)\rightarrow C(\mathbb{R}^d;\mathbb{R}^d)$ be defined by $C_g(f)\triangleq f\circ g$ for some fixed $g \in C(\mathbb{R}^d;\mathbb{R}^d)$.
What are examples of ...
- 5,405
2
votes
1
answer
280
views
Set operations over iterated function systems
An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,
$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$
is an IFS if each $...
- 21
3
votes
0
answers
94
views
How much more cyclic vectors are there than hypercylic vectors?
$\DeclareMathOperator\C{C}\DeclareMathOperator\HC{HC}$Definitions:
Let $T:X\rightarrow X$ be a bonded linear operator on a separable (infinite-dimensional) Banach space and define the sets:
$
\HC(T)\...
- 5,405
2
votes
1
answer
76
views
Hypercyclic vector for backshift operator
It is well-known that the weighted backshift operator $B_{\lambda}:\ell^p \rightarrow \ell^p$ is hypercyclic (with $\lambda>1$); that is, there exists a dense set of sequences $X\subseteq \ell^p$ ...
- 5,405
1
vote
1
answer
124
views
Cyclic vectors of translation operator
Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator
$$
t_a(f)\triangleq f(x)\mapsto f(x+a),
$$
is ...
- 5,405
2
votes
0
answers
165
views
Run-away Volterra operator
For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
- 5,405
0
votes
0
answers
221
views
Existence of the eigenvalue of the dual operator of the transfer operator
In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...
- 677
2
votes
1
answer
145
views
Orbit-based metric
Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric
$$
D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));
$$
...
- 5,405
2
votes
1
answer
137
views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
user121558
1
vote
0
answers
86
views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
2
votes
2
answers
375
views
Ergodic theorem and products
If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...
user135480
1
vote
0
answers
53
views
Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
- 1,043
2
votes
0
answers
293
views
Average of irrational flow on the torus
Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$
defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that
$$ \lim_{T\...
- 375
0
votes
0
answers
324
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
2
votes
2
answers
460
views
A possible dynamical approach to the "Invariant Subspace Problem"
In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...
- 356
6
votes
0
answers
281
views
Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
- 318
1
vote
1
answer
160
views
When will the $G$-invariant measure space be isomophic to the tracial state space of the crossed product $C^\ast$-algebra
Suppose a countable discrete amenable group $G$ acts continuously on a infinite Compact Hausdorff space $X$, i.e. $\alpha:G\curvearrowright X$. Suppose $\alpha$ is minimal. Write $M_G(X)$ for all $G$-...
- 181
1
vote
0
answers
336
views
Existence of solution for Poisson equation in Markov chain
Consider $X_n\in \mathcal{X}$ a controlled Markov chain taking value in a compact set $\mathcal{X}$ with action $a\in \mathcal{A}$, where the action set $|\mathcal{A}|$ is finite.
(In particular, we ...
- 185
6
votes
1
answer
469
views
Poincare Recurrence by Mean Ergodic Theorem
I have a question regarding a confusion from reading the Princeton Companion to Mathematics on the topic of Ergodics Theorems. It is about proving a stronger version of Poincare Recurrence Theorem ...
- 1,245
0
votes
2
answers
108
views
Density of positive orbits of $C_0$-semigroup
Suppose that $(T(t))_{t\geq0}$ is a $C_0$-semigroup on a Banach space $X$ and assume that there exists $x\in X$ such that $\{T(t)x:\ t\geq0\}$ is dense in $X$. I wonder why the set $\{T(t)x:\ t\geq ...
3
votes
1
answer
273
views
References on nonlinear evolution equations treated as infinite-dimensional systems for nonexperts
In many cases of interest a nonlinear evolution partial differential equation can be written as an infinite-dimensional dynamical system
$$
du/dt+A(t)u=0
$$
on a suitable functional space $X$, where $...
- 312
2
votes
0
answers
77
views
When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
12
votes
2
answers
811
views
Nuclear operators/spaces and transfer operators
While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
- 293
7
votes
2
answers
208
views
Characterizing when matrices are 'dissipative'
An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
- 291
10
votes
2
answers
1k
views
The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem
Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, $\mathbb{R}^{n}_+...
user11178
2
votes
0
answers
153
views
Size of the eigenfunction of Laplacian (reference request)
It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
$$||\phi||_{L^\...
- 1,692
3
votes
0
answers
113
views
Stationarity of Brownian motion with drift
Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...
- 165
5
votes
0
answers
237
views
Is Akcoglu's theorem for power bounded positive operators still an open problem?
I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...
- 1,594
1
vote
0
answers
318
views
Properties of a function from its pullback
Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{...
- 61
4
votes
1
answer
2k
views
The space of diffeomorphisms on a manifold
It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions (...
- 43
0
votes
1
answer
368
views
A variation of the Banach fixed-point theorem
Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...
7
votes
1
answer
319
views
Why aren't operator semigroups studied from a dynamical perspective?
Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying ...
- 277
28
votes
2
answers
2k
views
Dynamical properties of injective continuous functions on $\mathbb{R}^d$
Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all $x\in\...
- 419
5
votes
0
answers
221
views
Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
5
votes
2
answers
434
views
Sz.-Nagy dilation for uniformly convex Banach spaces
The Sz.-Nagy dilation theorem says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such that for all $x \in ...
- 6,287
1
vote
1
answer
172
views
Extension of power bounded operators over a finite subspace
Suppose $Y$ is a Banach space and $X$ is a finite-dimensional subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above and below, in other words ...
- 6,287
1
vote
1
answer
318
views
Integration by parts wrt. a Morse function on its basin of attraction
Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$
$$
\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...
- 53
3
votes
1
answer
445
views
Bohr sets, Coin-flip sets and Roth's theorem
I have been learning about Roth's theorem, trying to understand how Fourier series and dynamical systems (or even graph theory and binary sequences)are involved in counting arithmetic sequences in ...
- 22.8k
4
votes
4
answers
1k
views
Continuous pointwise ergodic theorem?
Let $\Phi$ be a homeomorphism of a compact metric space $M$
which preserves a regular Borel
probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. )
Under these hypothesis, I have two ...
- 8,336
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
47
votes
6
answers
6k
views
Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
- 15.5k
0
votes
1
answer
2k
views
Infinite linear span vs closed linear span
Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an ...
- 1,769
3
votes
2
answers
409
views
Convergence rate of an iterative process
I have the following iterative process
$$a_n=a_{n-1}(1-\phi(a_{n-1})),\quad 0< a_0<1,$$
where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< \phi(x)&...
- 931
1
vote
2
answers
659
views
A function which belongs on a concrete Besov Space
Please, anyone of you know a simple example of a function which belongs to the Besov Space with $p=q=\infty$ and $s=0$ (over $\mathbb{R}$ or $I\subset\mathbb{R}$ where $I$ is a closed interval). I ...
- 11