Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,399
questions
3
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Does Bernoulli imply exponential mixing?
This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
1
vote
1
answer
69
views
Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
1
vote
0
answers
94
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Question about ergodic flows and periodicity
Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is
continuous ...
- 729
1
vote
1
answer
116
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Does every proximal dynamical system have zero topological entropy?
A dynamical system is proximal if $$\:\forall (x,y) \in X \times X, \: \liminf_{n \rightarrow \infty} d(f^{n}(x),f^{n}(y)) = 0 $$ (where $X$ is a compact metric space with metric $d$). Is it true that ...
2
votes
0
answers
135
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Unipotent closure in classical groups
Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
- 21
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4
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791
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Ergodic theory applied to number theory
I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
0
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0
answers
49
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Sufficient conditions for chain recurrent set equal to set of non wandering points
Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
- 101
3
votes
1
answer
72
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Rate of convergence for Markov chain in random environment
Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic ...
- 131
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votes
0
answers
218
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Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
- 329
2
votes
1
answer
271
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When does uniqueness of a stable equilibrium imply it is globally stable?
Given a gradient dynamical system
$$\dot x=-\nabla f(x),$$
my question is:
(1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...
- 329
1
vote
0
answers
74
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Closed subgroups in Ratner's orbit closure theorem on unipotent flows
Let $G$ be a semisimple (real or $p$-adic) Lie group and $\Gamma$ a discrete and cocompact subgroup of $G$, as in the setting of Ratner's theorems on unipotent flows (see for example here \url{https://...
- 81
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48
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Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
0
votes
0
answers
25
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Coefficients in the series expansion of a central manifold are all zero
I have a system of 4 ODEs, which linearized around the origin gives
$$
\begin{align}
&\dot{q_1}=a\, q_1\\
&\dot{q_2}=b\,q_2\\
&\dot{q_3}=0\\
&\dot{q_4}=c\,q_4
\end{align}
$$
with $a$, $...
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1
answer
134
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Topological amenability of actions - forgetting topology
Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
- 516
3
votes
1
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162
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Chain components and posets
Let $(X,f)$ be a topological dynamical system ($f$ continuous, $X$ compact, metric with distance $d$).
Let $C\subseteq X^2$ indicate the chain recurrence relation:
$$xCy\iff \forall \epsilon>0\ \...
- 4,263
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votes
1
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153
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Help in understanding the singular system of linear forms and non escape of mass
I am having some trouble in understanding certain portions of the following paper by KKLM
https://link.springer.com/article/10.1007/s11854-017-0033-4
So in proposition 3.1, they proved the estimate ...
- 307
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votes
0
answers
55
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Diophantine-like approximation of dynamical subsystems
For $\alpha\in [0,1)$ irrational we know that there exists a sequences $\{ q_n \}_{n=1}^\infty\subseteq \mathbb{N}$ and $\{ p_n \}_{n=1}^\infty\subseteq \mathbb{Z}$ such that
$$ \Big\vert \alpha-\frac{...
- 579
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0
answers
33
views
Existence of a minimal, weakly mixing and Lipschitz selfmap?
I am looking for an example of a dynamical system $(M,f)$ such that:
$M$ is a metric space;
$f:M \to M$ is Lipschitz;
$f$ is weakly mixing (that is $f \times f$ is topologically transitive)
$f$ is ...
- 1
2
votes
0
answers
62
views
Name for a product of actions / dynamical systems
Suppose $G \curvearrowright X, H \curvearrowright Y$ are group (or monoid) actions, or dynamical systems. Then $X \times Y$ is a $G \times H$-system of the same type in the obvious way by $(g, h) \...
- 6,337
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88
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How to show that the map $ R $ here is measure-preserving
Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...
- 1,133
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Weak mixing and entering time
Let $X$ be a compact metric space and $f$ a continuous map from $X$ to $X$. Is it true, that if $f$ is weakly mixing, then the entering time $$N(U,V) = \{n \in \mathbb{N}\mid f^n(U) \cap V \neq \...
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155
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Has this metric been considered anywhere?
I posted this on math stack exchange some 10 days ago, but received no answers (https://math.stackexchange.com/q/4773194/1223994).
Let $X$ be a compact metric space and denote by $d$ the metric on $X$....
- 309
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0
answers
45
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Does a substitution tiling being FLC depend on starting seed?
I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
- 579
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votes
1
answer
112
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Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
- 109
1
vote
1
answer
255
views
Alternate definitions of compact and weak mixing extensions
In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition ...
- 4,832
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votes
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Understanding the application of two inequalities?
I am reading the paper "The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps" by Driss Kiouach and Yassine Sabbar.
I have two ...
- 185
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vote
1
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153
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Gradient descent under the presence of symmetries
Let $M$ be a Riemannian manifold (I'm happy to assume it is Euclidean space) with a function $f: M \to \mathbb R$ and a group of isometries $G$ acting on $M$ and preserving $f$, i.e., $f(gm) = f(m)$ ...
- 7,646
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answers
1k
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Continuous dynamical systems and analytic number theory: connections
Are there any connections between continuous dynamical systems and (analytic) number theory?
3
votes
2
answers
386
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Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 3,801
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Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
- 1,065
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votes
1
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Is the right-hand term of the autonomous dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, and ...
- 109
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votes
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Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]
Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$
i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$
Now let $X$ be the restriction of its ...
0
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2
answers
68
views
Is the right-hand term of the dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x,t),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z,t),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, ...
- 109
1
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0
answers
63
views
Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
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0
answers
239
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Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
- 312
1
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0
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147
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The space of ergodic elements of a topological or Lie group
Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
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The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
- 47
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votes
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answers
150
views
Local dimension of stationary measures for iterated function systems with an expanding map
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P = (p/2,p/2,1-p),$ where:
$f_1,f_2: I\to I$, where $...
- 233
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55
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Distortion lemma for composition of (distinct) functions expanding on average
I am trying to describe the following dynamics:
Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies:
$\forall \, \rho \in [0, 1], \, \exists \,...
7
votes
1
answer
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If the pointwise ergodic theorem holds along all subsequences with nonzero natural density, is the system strong mixing?
Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers whose natural density exists ...
- 4,832
0
votes
0
answers
50
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
1
vote
0
answers
161
views
Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...
- 312
1
vote
0
answers
172
views
Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
- 312
0
votes
0
answers
38
views
Generic non-existence of 1. Integral of continuous DS
Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by
$$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$
In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have ...
- 247
5
votes
1
answer
186
views
Bias of DS literature to polynomial ODEs
In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE
$$\...
- 247
2
votes
3
answers
342
views
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
3
votes
1
answer
294
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Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...
- 311
4
votes
0
answers
173
views
References for derivative w.r.t. initial condition of an ODE
Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have
$$
\sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
- 851
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votes
0
answers
296
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Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 247
0
votes
0
answers
46
views
A direct proof for non-zero limit points of weighted backward shifts
Fix a sequence $(w_1,w_2,\ldots)$ of positive reals such that the linear operator $T: \ell_2\to \ell_2$ given by
$$
T(x_1,x_2,x_3,....)=(w_2x_2,w_3x_3,\ldots) \text{ for all sequences in } \ell_2
$$
...
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