Questions tagged [discrete-dynamical-systems]

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2answers
209 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 ...
2
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0answers
38 views

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 $x \in \mathbb{R}^d$. ...
1
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1answer
87 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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0answers
58 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: ...
3
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0answers
64 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 ...
2
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1answer
68 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 ...
5
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1answer
144 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 ...
0
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1answer
69 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 ...
5
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0answers
81 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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0answers
106 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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0answers
47 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 $C^*(X,\alpha)$...
2
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1answer
132 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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0answers
85 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 ...
5
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1answer
234 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 $...
5
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0answers
166 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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0answers
22 views

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 ...
8
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3answers
224 views

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 ...
5
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1answer
451 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 ...
5
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0answers
145 views

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 ...
4
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0answers
91 views

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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0answers
70 views

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 ...
1
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1answer
178 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 ...
5
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2answers
117 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 ...
3
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0answers
53 views

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 $k \geq 1$. ...
23
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2answers
1k views

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})...