Questions tagged [discrete-dynamical-systems]

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76 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?
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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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108 views

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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1answer
82 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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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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1answer
538 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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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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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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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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1answer
76 views

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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2answers
186 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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575 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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81 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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1answer
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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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1answer
84 views

$\ell^1$-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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1answer
122 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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1answer
365 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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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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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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152 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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1answer
267 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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1answer
259 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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276 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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1answer
90 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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65 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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67 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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1answer
76 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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1answer
152 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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1answer
72 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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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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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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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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1answer
136 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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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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1answer
259 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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1answer
248 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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Practical statistics for queueing networks

There is a theory for queueing networks where we postulate some nicely behaving base distributions of arrival processes and service processes and then calculate the behaviour of the system. Now, in ...
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Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
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1answer
560 views

Orbits of the function f(x)=2x (mod 1)

I am currently studying the dynamics associated with the function $f(x)=2x$ (mod 1). In particular, if we define the orbit of an element $y \in [0,1]$ $$ orb(y)= \{ f^m(y): m \in \mathbb{Z}\}$$ it ...
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Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
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Percolation in torus under threshold rule

As part of my graduate research I am currently studying the last section in the paper "Random Majority Percolation" by Balister, Bollobas et al. The paper itself is very complicated but the last two ...
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Topological transitivity for a self-map of $\mathbb{R}$ with finitely many discontinuities

I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in ...
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1answer
181 views

Why do finitely many cluster variables imply finitely many y-variables?

Suppose we have a seed $(x,y,B)$ where $B$ is a skew-symmetrizable matrix, $x = \{x_1,\ldots,x_n\}$ and y is an n-tuple of elements in $Trop\{x_{n+1},\ldots,x_m\}$. If there are finitely many cluster ...
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2answers
123 views

General term formulas for nonlinear recurrence sequences

It seems to be a well known question: in which cases will there be general term formulas for sequences like $p_n=a p_{n-1} ^2 +b p_{n-1} +c$ where $a, b, c$ are real or complex numbers and n is ...
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Self-map of a set for which the sizes of fibers of iterates are given by polynomials

I am interested in functions $f\colon X\to X$ (where $X$ is some countable set) such that for every $x \in X$ there exists a polynomial $P_x$ such that $\#(f^k)^{-1}(x)=P_x(k)$ for all $k \geq 1$. ...
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Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $f^{n}(x_{0})...