# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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questions

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### Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...

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### Time-scale calculus (an similar approaches - measure chains) on more general "time" sets

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...

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1
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### Lyapunov exponents of convergent sequence of matrices

Let $A_n$ be a sequence of $d \times d$ matrices converging to a matrix $A$, all invertible and diagonalizable. We can define the Lyapunov spectrum of the corresponding dynamical system:
$$ \chi = \...

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### Construction of a homogeneous Moran set

Fix a positive integer $N\ge 2$, for $n \in \mathbb{N}$, denote
$$\Sigma=\{0,1,\dots,N-1\},\\
\Sigma^n=\{(\omega_1,\dots,\omega_n):\omega\in\Sigma, i=1,\dots,n\}.$$
Let $p>2$ be a positive integer. ...

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### Hopf normal form of 2D system – How to calculate parameter $\beta$?

In the Scholarpedia article on the Hopf bifurcation, it is discussed that any generic (nondegeneracy and transversality) 2D system exhibiting a Hopf bifurcation has a locally topologically equivalent ...

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### Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/\operatorname{SL}(d,\mathbb Z)$ is ergodic

I wonder if there are any direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/\operatorname{SL}(d,\mathbb Z)$ is ergodic (or even stronger, mixing). ...

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### Does locally nilpotent imply nilpotent for continuous self-maps of intervals?

Let $f\in C([0,1],[0,1])$ be such that:
$$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$
Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)...

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### Ergodic diffeomorphisms of the circle

From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...

4
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1
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291
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### A process of repeated convolution and conditioning and the resulting sequence of probability distributions

I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$.
Let $D_1$ be the $n$-dimensional Gaussian distribution with ...

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### Reference for rigidity of higher rank action

I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...

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### Smoothness of unstable manifold near (non?)-hyperbolic fixed point w.r.t. generator of the flow

Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can ...

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### Properties of the quadratic form for the sum of the Jacobian and its transpose

Consider the Jacobian matrix of
$$
\begin{bmatrix}
f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\
\vdots \\
f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))
\end{bmatrix}
$$
Let the jacobian ...

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1
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### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...

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### Regularity of center manifold

Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...

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### A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...

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0
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### Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...

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### Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...

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### Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?

The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...

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1
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### Examples of ODEs with complex constant coefficients and applications to physics?

This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics?
but received no answers. I am reposting it here on the hope that it catches ...

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0
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### Weakly mixing diffeomorphism

From
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...

6
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2
answers

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### Poincaré recurrence and its implications for statistical physics and the arrow of time

A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...

4
votes

1
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214
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### What is the name for a point that is periodic to within $\varepsilon$?

Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...

2
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### Examples of chaotic self-similar blowup in PDEs

When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference:
Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in ...

6
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1
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395
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### Equidistribution modulo 1

We know that the time spent by the sequence $na \mod 1$, $n$ ranging from $1$ up to $x$ and $a$ irrational, at any interval of length $\delta$ is approximately $\delta x$. There are known results when ...

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### Stable sets for gradient flow of functions with singularities

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a real-analytic function, and let $F_t$ denote the gradient flow of $f$ with respect to some background metric. Suppose that $df = 0$ at a point $p$. In the ...

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1
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### Local rule for the product of two cellular automata

Consider two one-dimensional cellular automata $(A^{\mathbb Z},F)$ and $(B^{\mathbb Z},G)$ with alphabets $A$ and $B$ and global rules $F: A^{\mathbb Z} \to A^{\mathbb Z}$ and $G: B^{\mathbb Z} \to B^{...

6
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### Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof

I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...

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### The Logistic map have subexponential decay of correlation?

I was looking for information about the correlation decay of the logistic map, more precisely if there is any parameter for which its decay is subexponential, in which case I would like to know if it ...

2
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### Search for period N logistic map

The logistic map is a period doubling bifurcation system.
Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...

3
votes

1
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### Solution to a Sylvester equation with positive definite coefficients

Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$
\begin{align*}
C = A^TXA + B^TXB.
...

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0
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### How would you approach unknown states/variables in steady-state or equilibrium?

Consider a linear state space system $ \dot{x} = Ax + B$, with $x$ being a $n\times 1$ vector of $n$ state variables, and $A$ and $B$ being known matrices with dimensions $n\times n$ and $n\times m$, ...

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### Entropy of a sequence

I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,
It would be interesting to rule out the possibility of $\lambda_f(n)$ ...

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0
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### What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?

Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...

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0
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### Oseledets splitting of $f_{0}$ is either dominated with index 1 or trivial at $x$

There is a well-known result by Bochi and Viana that said
Let $f_{0} \in \operatorname{Diff}_{\mu}^{1}(M)$ be such that the map
$$
f \in \operatorname{Diff}_{\mu}^{1}(M) \mapsto\left(\mathrm{LE}_{1}(...

4
votes

2
answers

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### Borel summation and the Abel function of $e^z-1$

This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...

6
votes

1
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### Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...

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0
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### Lipschitz continuous paths on manifolds which accumulate on non-injective paths

This question is concerned with a dynamical system given by a shift flow on a metrisable topological space of functions from $\mathbb{R}$ to a closed manifold. The choice of manifold affects the ...

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0
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### Basin of attraction comparative statics* using local energy functions?

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...

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### Uniformly weak mixing transformations

Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is uniformly weak mixing if for every $\varepsilon > 0$, ...

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0
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### Denjoy example in the Poincaré–Bendixson theorem

I have already finished understanding the Poincaré-Bendixson theorem as a consequence of Schwartz's theorem, but I also want to analyze the example that Denjoy gave in $C^1$ that is not within the ...

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### Naturality of geodesic flow

Let $\texttt{Man}$ be the category of smooth manifolds with local diffeomorphisms as morphisms, and $ \texttt{Bun}$ --- the category of bundles (affine bundles or just fibre bundles, if necessary) and ...

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1
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### Analytically characterizing basins of attraction boundaries and sizes

While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...

6
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2
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214
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### Topological dynamical systems with only zero-entropy factors

Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?

44
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### Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of ...

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### Motivation for Ionescu Tulcea-Marinescu (Lasota-Yorke inequality)

I wonder about motivations of a work of Ionescu Tulcea-Marinescu. In order to establish the decomposition of the operator $T$ they assume (condition (1.3)) this operator satisfies the inequality
$$\|...

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1
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### Does uniform recurrence imply uniform convergence of the Birkhoff sums?

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure.
Suppose $T$ is uniformly recurrent, in the following sense:
For every $A \in \mathcal F$, there exists an $M \...

4
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1
answer

132
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### Ergodic decomposition of the action of a subgroup

Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the ...

3
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1
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### the definition of the topological pressure for matrices

Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...

2
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1
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### Finding the repelling fixed point of an exponential, knowing only its attracting one

This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...

3
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### What dynamical properties should we expect from systems satisfying statistical ones?

Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...