# 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 ...
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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1 vote
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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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### 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
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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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1 vote
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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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### 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 ...
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 ...
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 ... 46 views

### 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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### 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 ...
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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### 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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### 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 $x \in \mathbb{R}^d$. ...
1 vote
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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### 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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### 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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### 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 ...
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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### 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 ...
$\DeclareMathOperator\C{C}\DeclareMathOperator\HC{HC}$Definitions: Let $T:X\rightarrow X$ be a bonded linear operator on a separable (infinite-dimensional) Banach space and define the sets: $\HC(T)\... 4 votes 0 answers 110 views ### Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra Let$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)$... 3 votes 0 answers 50 views ### The number of minimal components of a dynamical system via certain invariants of corresponding cross product$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)$... 2 votes 1 answer 138 views ### Orbit-based metric Let$(X,d)$be a metric space and$f:X\rightarrow X$be continuous. Then, is there any meaning/research done on the metric $$D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));$$ ... 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 ... 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$...
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 \...