Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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What is the asymptotic dynamics of the winning position in this game?
$n$ players indexed $1,2,...,n$ play a game of mock duel. The rules are simple: starting from player $1$, each player takes turns to act in the order $1,2,...,n,1,2,...$. In his turn, a player ...
- 2,619
3
votes
1
answer
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views
Exact solution to a periodic linear ODE sought
We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
- 33
3
votes
1
answer
229
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Existence of topologically transitive map on Euclidean space
I was reading this post and wondered. Does there exist a topologically transitive (TT) map $f:\mathbb{R}^n\to\mathbb{R}^n$ when $n\geq 2$? I know that post asks for compactness and topological ...
- 5,405
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0
answers
30
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Hypercylic operators with sets of hypercyclic vectors almost covering the space
Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in ...
- 5,405
1
vote
1
answer
127
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Extension of the definition of entropy to $\mathbb{Z}^d$ and $\mathbb{N}^d$
I read the paper Entropie d'un groupe abélien de transformation by Conze and the part of the book Dynamical systems of Algebraic Origin by Schmidt about the entropy for $\mathbb{Z}^d$ actions. I was ...
- 121
1
vote
3
answers
435
views
Operator norm of shift operator for finite measure spaces
Let $\nu$ be a finite Borel measure on $\mathbb{R}^n$ and define the shift operator $T_a$ on $L^p_{\nu}(\mathbb{R}^n)$ by $f\to f(x+a)$ for some fixed $a\in \mathbb{R}^n-\{0\}$. Suppose moreover that ...
- 5,405
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votes
0
answers
104
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Simulating measurement data from a dynamical system and its derivative [closed]
I am implementing the SINDy algorithm in Python.
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
I have a question concerning the simulation of ...
- 21
2
votes
1
answer
160
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Dynamical system described by coupled nonlinear differential equations
Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...
- 197
0
votes
1
answer
409
views
von Neumann ergodic theorem for $L_p$
Let $\tau:\Omega\to \Omega$ be a measure-preserving transformation with $\mu(\Omega)<\infty$. Define $T:L_p(\Omega)\to L_p(\Omega)$ as $Tf:=f\circ \tau$. I want to prove that for all $1\leq p<\...
- 1,879
5
votes
2
answers
497
views
Classification of the behaviours of the logistic map
On this this wikipedia page, it is claimed that the iterative sequence $x_{n+1}=rx_n(1-x_n)$ (the logistic map) starting at a point $[0,1]$ and where $r$ ranges in $[0,4]$ behaves differently ...
- 489
0
votes
0
answers
369
views
Definition of generic point
I am trying to read a paper named D.S. Ornstein, B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math. 79 (1) (1992) 113–127, doi:10.1007/BF02764805. In this paper the authors ...
- 1,879
3
votes
0
answers
78
views
Example of primitive substitution with two rationally independent eigenvalues?
I am looking for an example of a primitive substitution $\sigma$, not Pisot, such that the associated subshift $X_\sigma$ has two irrational and rationally independent eigenvalues. Equivalently, a ...
- 31
7
votes
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157
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Non-compact dynamical systems
In topological dynamics, most of the time, we consider the continuous action of a (semi)group $G$ on a compact Hausdorff space $X$. In this context, we can envelop the group in a compact left ...
- 1,338
5
votes
1
answer
313
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Smoothening pseudo-Anosov flows
A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...
- 235
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0
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271
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Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?
The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base $e:=\exp(1)$.
Update The question is a complete rewriting ...
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votes
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475
views
If $(Y,T)$ is a connected minimal system with a symbolic extension of linear word complexity, is $(Y,T)$ equicontinuous?
Let $(Y,S)$ be a minimal topological dynamical system such that $Y$ is connected. A simple example of a system like this is an irrational rotation of the circle, and it is known that Sturmian ...
- 151
1
vote
0
answers
52
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On smooth extensions of functions
Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...
- 105
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votes
1
answer
116
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is the Lyapunov exponent a continuous function of the invariant measure w.r.t weak-* topology?
I hope for some relerant results for the following question:
Is the lyapunov exponent continuous with respect to the measure?
Assume $M$ is a manifold, $f$ is a diffeomorphism on $M$, $m$ is an ...
- 51
2
votes
2
answers
241
views
Transverse invariant measures to vector fields
Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure?
I've seen examples of higher-...
- 1,601
4
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0
answers
110
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Counting simple closed curves
I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
- 4,418
2
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0
answers
83
views
Center-stable manifold theorem on manifold with boundary
I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
- 21
0
votes
1
answer
46
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Difference of hypercyclic operator and identity
Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I_B$ the identity on $B$, and define $L_k: =k (I_B - L)$. When is $L_k$ hypercyclic on $B$? Can anything else ...
- 5,405
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votes
0
answers
184
views
Question on "semi-linear" dynamical systems
I am interested in understanding the convergence properties of a dynamical system at zero. We know that a dynamical system of the form $x_{k+1} = Ax_{k}$ where $A$ is a symmetric positive definite ...
- 105
3
votes
2
answers
137
views
Does this strong form of being almost 1-to-1 imply injectivity?
Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that
$\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
$\tilde{Y}$ is a ...
4
votes
1
answer
317
views
Properties of the spectrum of the Koopman representation
Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an ...
- 111
1
vote
1
answer
125
views
Action of a toral automorphism on a Markov partition
Let $$A = \begin{bmatrix}
1 & 1 \\
1 & 0\\
\end{bmatrix}.$$
Then the eigenvalues of $A$ are $1/2(1+\sqrt{5})$ and $1/2(1-\sqrt{5})$. The eigenvector corresponding to the unstable eigenvalue ...
- 113
1
vote
0
answers
97
views
Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
4
votes
2
answers
332
views
Exponential iterates of a complex number
Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$.
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
- 3,317
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vote
0
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Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 5,405
5
votes
0
answers
183
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Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
5
votes
1
answer
179
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Physical measures that are not SRB
It is quite easy to construct a dynamical system which has a physical measure with a positive Lyapunov exponent and zero entropy, just a figure $\infty$ system. By Pesin's entropy formula such a ...
- 1,648
2
votes
2
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1k
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Monotonic and bounded sequences throughout mathematics [closed]
When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
- 4,943
5
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2
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297
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Seeking to understand meaning of "von Neumann spectrum" in a paper of Bader–Furman–Shaker
In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page)
I find that towards the end of the ...
- 2,125
1
vote
0
answers
67
views
Stability of a continuous piecewise linear map
I am studying random perturbation of a system that is continuous and piecewise linear. More precisely: I am given a map $\Phi_1:\mathbb{R}^d\to \mathbb{R}^d$ such that
$$ \Phi_1(x) = \left\{\begin{...
- 562
4
votes
1
answer
358
views
Let $U$ be a simply connected open subset of ${\Bbb S}^2$, is the complement of $U$ also simply connected?
I was looking into particular cases for the Poincaré-Bendixson theorem and I came across a topological problem about simply connectivity.
If $\gamma$ is a Jordan curve in ${\Bbb S}^2$ then using ...
- 462
8
votes
1
answer
493
views
Metrics on torus without closed contractible geodesics
It is easy to see that any closed geodesic on a flat 2-torus is noncontractible.
Further the same holds true for a torus of revolution.
Indeed either a closed geodesic is a meridian and therefore ...
- 45k
19
votes
2
answers
1k
views
Existence of continuous map on real numbers with dense orbit?
Does there exist a continuous map $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the forward orbit of 0 is dense in $\mathbb{R}$?
- 189
3
votes
1
answer
233
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A subadditive maximal ergodic theorem
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
- 167
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votes
2
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883
views
Do infinitely nested radicals have any applications?
There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
- 4,943
7
votes
2
answers
732
views
A second order nonlinear ODE
In my research (in differential geometry) I recently came across the following nonlinear second order ode:
$$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$
It ...
- 277
1
vote
0
answers
68
views
Alternate characterization of floquet multipliers: Floquet theory
Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. ...
- 11
26
votes
4
answers
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...
- 385
3
votes
1
answer
121
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Almost one-to-one endomorphism of minimal subshift?
Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between ...
4
votes
1
answer
310
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Leaves of stable foliation of holomorphic Anosov diffeomorphism
I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...
- 143
11
votes
0
answers
212
views
Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)
Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...
- 695
2
votes
0
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158
views
Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$
I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\cdot \big(v,\,w\big)\...
- 203
1
vote
0
answers
64
views
Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
0
votes
0
answers
55
views
Dense stratification of a separable Hilbert space
Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
- 5,405
1
vote
0
answers
89
views
Lower bounds of kappa class functions
I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of
Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R}
^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any ...
- 61
0
votes
1
answer
134
views
Consequences of invariant-subspace problem to Li–Yorke chaos [closed]
The invariant-subspace problem is probably an open problem for reflexive spaces which asks:
Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...
