Questions tagged [discrete-dynamical-systems]

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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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4 votes
2 answers
221 views

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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6 votes
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445 views

Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below). Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...
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1 answer
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Why all the coefficients of the center manifold of this system are zeros?

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
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1 vote
1 answer
149 views

A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products

Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...
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3 votes
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Description of Anderson-Putnam CW-complex construction

I have been trying to read the paper, Topological invariants fo substitution tiling and their associated $C^*$-algebras, to learn more about a construction of Anderson-Putnam complexes. However, it ...
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$L^p$-continuity for discrete linear causal systems

Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by: \begin{...
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Explicit bound computation for Kalman filter covariance $\|P_{k|k}\|$ as a function of $\|P_0\|$

My Goal: Given and initial covariance $P_0$, a Kalman filter updates covariance $P_{k|k}$ according to a nonlinear update equation. I am looking for a bound $B=B(\|P_0\|)$ such that $\|P_{k|k}\|\leq ...
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1 vote
1 answer
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Invariant distributions for iterated random variables (stochastic dynamical systems)

This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...
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1 answer
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Do measure-valued dynamical systems correspond to marginals of Markov processes?

Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
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Hypercylic operators have very typical cyclic vectors

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}...
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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 $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....
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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?
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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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Gradient-like dynamical systems

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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  • 141
2 votes
1 answer
106 views

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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2 votes
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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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11 votes
1 answer
639 views

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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124 views

Example of topologically transitive dynamical system with invariant non-ergodic Borel measure

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 \...
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  • 141
3 votes
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122 views

Is composition of discrete Hamiltonian flows integrable?

Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$ For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
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24 votes
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Do these rational sequences always reach an integer?

This post comes from the suggestion of Joel Moreira in a comment on An alternative to continued fraction and applications (itself inspired by the Numberphile video 2.920050977316 and Fridman, ...
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1 answer
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Why does bounded distortion imply the following inequality?

Let $f: I \to I$ be a one-dimensional differentiable function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such ...
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3 votes
2 answers
193 views

Showing that the inverse of a function is approximately equivalent to $\frac{1}{n^{1/\alpha}}$

I'm currently working with someone on my PhD, and last week they asked me to check that a certain approximation holds as an exercise. Unfortunately, I couldn't figure out how to do it, and we've since ...
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2 votes
3 answers
613 views

The critical exponent function

It is a known fact [1] that, for every $c\in (1,\infty]$, it is possible to find a finite alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has critical exponent $c$. It looks ...
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2 votes
0 answers
90 views

Rotation set vs existence of rotation number

Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...
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4 votes
1 answer
74 views

On composite divisors of certain terms in the extended Lucas sequences

One can easily show that if $p$ is a prime that does not divide $a$, then $a^{p(p-1)}\equiv 1 \pmod{p^2}$. However, my question is: If instead of $p$ being a prime, it were a pseudoprime to the base $...
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1 vote
1 answer
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$\ell^1$-bound on graph laplacian with weight

Consider the $\mathbb Z^2$ lattice, we then define for $u=(u_{ij})_{i,j \in \mathbb Z}$ the discrete Laplacian $$(\Delta u)_{i,j}=u_{i+1,j}+u_{i-1,j}+ u_{i,j+1}+u_{i,j-1}$$ and the weight which pushes ...
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4 votes
1 answer
182 views

Maximal length of trajectories in billiard

Consider discrete rectangular billard on lattice with integer dimensions a*b and n balls with radius $\frac{\sqrt 2}{2}$ and ...
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4 votes
1 answer
423 views

Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form $$ x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)} $$ for ...
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1 vote
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Property of a map on the 2D lattice

My brother ($A_1$) and I ($A_2$) are playing together with the following "game". My brother picks a positive integer $m_1$ and then draw a line of length $m_1$ (units). I pick independently ...
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3 votes
0 answers
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Convergence of iteration of a convex program

Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...
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2 votes
0 answers
158 views

Permutations with bounded displacement on a circle

I am interested in references to results on permutations $\sigma$ of $\{0,\ldots,n-1\}$ satisfying $\text{min}\{ \sigma(i)-i\text{ (mod } n),i-\sigma(i) \text{ (mod } n) \} \leq k$ for some $k$ for ...
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  • 323
3 votes
1 answer
294 views

Run-away functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. We say that f has the run-away property if for every compact subset $K\subseteq \mathbb{R}$ there is some positive integer N such ...
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2 votes
1 answer
303 views

concentration inequality for product of matrices

Suppose that we have $N$, $n\times n$ positive semidefinite matrices $A_1, \cdots,A_N$ such that $0\preceq A_i \preceq I_n$, $\forall i$. Also let us assume that $A_i\neq I_n$ for all $i$ and they are ...
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4 votes
2 answers
308 views

A possibly easy question about latent geometry in Collatz sequences

I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be aware ...
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Probabilistic Approximation of non-linear Dynamical System by Diffusion Process

Setting Suppose I have a discrete dynamical system given by: $$ X^{n+1} = f(X^{n}) \qquad X^0 =x , $$ where $f$ is some diffeomorphism from $\mathbb{R}^{d}$ to itself, and some $x \in \mathbb{R}^d$. ...
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1 vote
1 answer
111 views

Continuous-time extension of a discrete dynamical system

It is clear that one can obtain a discrete dynamical system from a continuous one, but is the converse possible if the system is "nice"? Define the discrete-time dynamical system on $\mathbb{R}^d$ by ...
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0 votes
0 answers
69 views

For which parameters is the logistic map chaotic?

The logistic map is $f_\lambda(x)=\lambda x (1-x)$. It is known that the map is chaotic for $\lambda=4$ (on $[0;1]$) and also for $\lambda>0$ (on some hyperbolic subset of $[0,1]$). My question is: ...
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3 votes
0 answers
69 views

How often can we renormalize unimodal maps?

A unimodal map is function $f:[0,1]\rightarrow [0,1]$ such that there exist $c\in (0,1)$ such that $f$ is strictly increasing on $[0,c)$ and strictly decreasing on $(c,1]$. A unimodal map f is ...
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2 votes
1 answer
112 views

Fine structure of bifurcation diagram of logistic family

I'd like to learn about the period-doubling route to chaos of the logistic family $f_\lambda(x)= \lambda x (1-x)$ and got interested in the fine properties of the bifurcation diagram of this family as ...
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3 votes
1 answer
157 views

Reversal of open cover with topologically transitive dynamical system

Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel ...
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0 votes
1 answer
79 views

Topologically transitive dynamical system mapping space into ball

Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
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3 votes
0 answers
91 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)\...
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4 votes
0 answers
110 views

Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra

Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...
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3 votes
0 answers
50 views

The number of minimal components of a dynamical system via certain invariants of corresponding cross product $C^*$ algebra, some precise examples

Let $X$ be a compact Hausdorff space and $\alpha$ be a homeomorphism of $X$. So we have a natural action of $\mathbb{Z}$ on $C(X)$ which generates the cross product algebra $C^*(X,\alpha)$...
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2 votes
1 answer
138 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 $$ D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y)); $$ ...
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2 votes
0 answers
111 views

On invariant cones of the Katok map

I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
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  • 141
5 votes
1 answer
279 views

An entire function all whose forward orbits are bounded

Edit: I revise the question according to the comment of Gabe Conant. What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?: For every $...
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5 votes
1 answer
272 views

"Determinant" rather than "trace" in the alternative formula "Lefschetz number"

For a self map $f$ on a topological space $X$ we replace "trace" with "determinant" in the alternative Lefschetz formula $$\Lambda(f)=\sum(-1)^i trace(f^*)|H^i(X,\mathbb{Q})$$ So we have $$\...
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