# Questions tagged [discrete-dynamical-systems]

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### Asymptotic decay rate of an oscillator integral

Question: I want to evaluate the decay estimate of the integral $I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta$ for ...
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### Renormalization group in condensed physics and field theory

Renormalization group in field theory is differenct from the one in condensed physics in that the former satisfies a differential equation, but the latter does not. Does Renormalization group in ...
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### Estimate for the length of a partial orbit for a shift map for which its delta neighbourhood covers an interval

Consider $f:[0,2\pi) \to [0,2\pi )$ given by $f(x) = (x + 1) \bmod 2\pi$ for all $x\in [0,2\pi )$, i.e. a shift map on the unit circle with anti-clockwise shift of $1$. Denote the sequence $\{ x_n \}$ ...
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### Finite approximability of graphs with finitely many automorphisms

In this question, all graphs are understood to be simple and undirected, and have countably many vertices and edges, but not necessarily finite. Let $G = (V, E)$ be a graph. It is clear that any ...
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### Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map \begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned}, starting ...
1 vote
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### Does every proximal dynamical system have zero topological entropy?

A dynamical system is proximal if $$\:\forall (x,y) \in X \times X, \: \liminf_{n \rightarrow \infty} d(f^{n}(x),f^{n}(y)) = 0$$ (where $X$ is a compact metric space with metric $d$). Is it true that ...
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### How to show that the map $R$ here is measure-preserving

Assume that $(X,\mathcal{B},m,T)$ is a measure-preserving dynamical system, where $(X,\mathcal{B},m)$ is a probability space, $\mathcal{B}$ denotes all the measurable sets in $X$, $m$ is the ...
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### The boundedness of dynamical systems discretized from Hamiltonian systems

Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e., \begin{align} &\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\ &\frac{dq}{dt}...
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### How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?

Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...
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### How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
1 vote
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### Existence of center-stable manifold when the Jacobian is singular?

The following is a result from Shub's monograph "Global Stability of Dynamical Systems". I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...
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### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye : "If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
271 views

### Unbounded solution but bounded Euler discretization

Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
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### Is there a classical textbook/reference on numerical discretization schemes?

I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...
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### Topological dynamical systems with only zero-entropy factors

Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?
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### Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of ...
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### Reduced $H_{\infty}$ problem for nD systems

Let $G(z)$ denote the (rational not necessarily square and unstable) transfer function of an nD system, where $z=(z_{2},...,z_{n})$, of a discrete spatial-temporal recurrence Givone-Roesser type ...
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### Construction of minimal zero entropy measure-theoretically strong mixing subshift?

Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is (1) minimal (2) zero (topological) entropy (3) measure-theoretically strong mixing (for some measure)? I am in particular ...
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### Distortion estimates to control Hausdorff measure of a curve

I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent. I have a problem to understand how the distortion estimates are used. The ...
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### Is there half an iteration of the QR algorithm?

Every real square matrix $M$ has a QR decomposition $M = QR$ where $Q^{-1}=Q^T$ and $R$ is an upper triangular matrix with non-negative reals on the diagonal. Call the function $f(QR)=RQ$ the Francis ...
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Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1}... • 5,089 0 votes 0 answers 63 views ### a lemma on interval translation map Consider the map$S:T^1 \to T^1$where$x \mapsto x+c_j$, mod 1 where$c_j$'s are real numbers. We represent$T^1$as a union of disjoint subsegments$M_j=[t_j,t_{j+1})$,$j=0 , \cdots ,n , t_0=t_n$.... 1 vote 0 answers 195 views ### Is there a condition for a subshift of finite type to be uniquely ergodic? Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic? • 111 2 votes 0 answers 58 views ### Chain recurrent points of a gradient-like system Let$X$be a compact metric space and$f:X\to X $homeomorphism. Let$V:X\to \mathbb{R}$be a Lyapunov function for$(X,f)$(continuous function such that$(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...
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I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...
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### Busy beaver sequence for a simple tag-like system

This question arose in the context of tag-like systems, specifically Bitwise Cyclic Tag (BCT). Consider the following discrete dynamical system: Let $\mathbb{B} = \{\mathtt{0}, \mathtt{1}\}$. Let our ...
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### Is this variant of bitwise cyclic tag Turing-complete? [closed]

Cross-posted from Theoretical Computer Science. CT is an extremely minimalist programming language that can simulate arbitrary tag systems, and is therefore Turing-complete. A program consists simply ...
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### Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$. You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...
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Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which \$f : \Lambda \to \...