# Tagged Questions

**0**

votes

**1**answer

209 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 ...

**6**

votes

**1**answer

221 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 ...

**27**

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 ...

**5**

votes

**0**answers

169 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 ...

**4**

votes

**2**answers

269 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 ...

**1**

vote

**1**answer

117 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 ...

**1**

vote

**1**answer

209 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 ...

**2**

votes

**1**answer

291 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 ...

**2**

votes

**4**answers

343 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 ...

**0**

votes

**0**answers

140 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 ...

**8**

votes

**0**answers

123 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 ...

**27**

votes

**6**answers

2k 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 ...

**0**

votes

**1**answer

376 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 ...

**3**

votes

**2**answers

208 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< ...

**1**

vote

**2**answers

239 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 ...

**1**

vote

**1**answer

336 views

### semi group of contractions

Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative, and let $B$ is a monotone linear operator such that $D(A)\subset D(B)$.
...

**0**

votes

**1**answer

161 views

### Dissipative operator

Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.
is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?
Thank ...

**0**

votes

**1**answer

393 views

### Hölder continuity of uniform limit of piecewise constant functions

Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants ...

**1**

vote

**4**answers

307 views

### A Fractional Linear Transformation Class Property

Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and ...

**10**

votes

**1**answer

591 views

### A measure theory question

Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...

**3**

votes

**1**answer

271 views

### Stronger bound for a modified Lyapunov Equation

In regard to the stability analysis and control properties of the linear system $\dot{x}=Ax$.
Consider the solution $P$ of the continuous Lyapunov equation $AP+PA^T+Q=0$, where $A,Q,P
\in
...

**1**

vote

**0**answers

207 views

### Approximation of the radon-derivative

I am looking for the following statement.
Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by ...

**3**

votes

**1**answer

226 views

### Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators

A research problem on which I am currently working requires a construction in topological dynamics of the following type:
Let $T \colon X \to X$ be a continuous transformation of a compact metric ...

**10**

votes

**3**answers

795 views

### Alternative proofs of the Krylov-Bogolioubov theorem

The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...

**0**

votes

**1**answer

287 views

### When are operators extended by linearity bounded?

Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...

**1**

vote

**3**answers

291 views

### Almost Orthogonal Vectors given a Unitary Operator

Let $\mathit{H}$ be a (real or complex) Hilbert space and $U:\mathit{H}\rightarrow\mathit{H}$ be a unitary operator. What conditions can be placed on $U$ to guarantee a sequence $v_n$ such that ...

**1**

vote

**1**answer

215 views

### Shift operator that generates separable orbit

Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset ...

**8**

votes

**0**answers

355 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**1**

vote

**0**answers

204 views

### Variation of a function

There are probably some of you guys who already know some of the terms that I am going to use so in order to be not so boring I will put the definition to the end.
Let $f$ be a piecewise expanding ...

**14**

votes

**13**answers

3k views

### Is there a “crash-course” book on Abelian varieties (e.g., an introduction for physicists)?

Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...

**3**

votes

**1**answer

484 views

### Do maps have flows?

In A New Kind of Science: Open Problems and Projects(pg. 36).
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...

**2**

votes

**2**answers

762 views

### Do the Euler method's approximations always approach the true solution?

Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the ...

**7**

votes

**3**answers

364 views

### Noninteger iterates of functions: How to get ODE from flow at a given time?

Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, ...