0
votes
1answer
210 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
1answer
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
2answers
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
0answers
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
2answers
270 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
1answer
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
1answer
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
1answer
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
4answers
348 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
0answers
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
0answers
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
6answers
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
1answer
378 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
2answers
209 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
2answers
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
1answer
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
1answer
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
1answer
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
4answers
308 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
1answer
592 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
1answer
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
0answers
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
1answer
227 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
3answers
797 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
1answer
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
3answers
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
1answer
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
0answers
356 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
0answers
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
13answers
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
1answer
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
2answers
763 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
3answers
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, ...