# 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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### $L^p$-continuity for discrete linear causal systems

Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by: \begin{...
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### Equivalence between smoothly regular and analytically regular

I think the following statement is true. Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
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### Reference of the fact that Hoelder cocycles are associated to Hoelder potentials in Ledrappier's correspondence

Let $\tilde{M}$ be the universal cover of a compact pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. ...
31 views

### Statistical characteristics of low complexity subshifts

I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
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### The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system

I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum. For a compact ...
174 views

### Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?

Consider a continuous ODE, $$\dot x = f(x), f \in C^1$$ $\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
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### Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$

I'll write it formally: Let $\sin^{\circ\, 1}(x) = \sin(x)$ and $\sin^{\circ n+1}(x) = \sin\bigl(\sin^{\circ n}(x)\bigr)$ for $n\in \Bbb N$ with $n>1$. What is the limit as $n \to \infty$? It's ...
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### Open problems in symbolic dynamics

I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. ...
287 views

### Neural networks over gadgets other than $\mathbb{R}$

Recently, I learned that neural networks (NN) can be defined over fields other than $\mathbb{R}$: for example, Khrennikov and Tirozzi wrote a paper in 1999 (!) on $p$-adic neural networks, or neural ...
146 views

### What is a holomorphic foliation?

For a smooth foliation $F$, there are three equivalent definitions: the leaves of $F$ are tangent to a smooth vector field; the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
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### Existence of a continuous ergodic dynamical system for a given distribution?

It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...
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### What is the name of this bifurcation

I've seen a bifurcation occur in several textbooks where below the bifurcation point, there are zero fixed point, above the bifurcation point there is one fixed point, and at the bifurcation point ...
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### The mean ergodic theorem for weakly mixing extension

I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help. I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
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### Has the von Neumann entropy ever been used in classical mechanics?

After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience , it occurred to me that this entropy might have ...
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Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces. Consider the set $D_{m,n}$ of $m \times n$ real matrices satisfying that the system of inequalities \|Aq-p\|^m < \frac{1}{T}, \|q\|^... 1answer 147 views ### Intuition for almost periodic solution and Poincaré recurrence theorem I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer. Suppose that we have a PDE that admit a solution u that can be ... 0answers 70 views ### Ergodic action on product spaces Let (X_1 \times X_2,d\mu) be a measure space with X_2 compact. Suppose that we have a continuous (diagonal) action of a topological group G on X=X_1 \times X_2. I know that the action of G ... 1answer 217 views ### A property of rapid sequences of natural numbers \newcommand{\IR}{\mathbb R} \newcommand{\IT}{\mathbb T} \newcommand{\w}{\omega} \newcommand{\e}{\varepsilon} Taras Banakh and me proceed a long quest answering a question of ougao at ... 0answers 111 views ### Dynamical phenomena in \mathbb{R}^n first arising for n > 3? For differentiable dynamical systems defined on, say, an open ball in \mathbb{R}^n, when n=2 Poincaré-Bendixson tells us a lot about what can happen. In particular, P-B precludes chaos and strange ... 0answers 31 views ### Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable? Let \Omega and M be compact C^\infty manifolds, let \theta \colon \Omega \to \Omega be a C^\infty diffeomorphism, and let \Theta \colon \Omega \times M \to \Omega \times M be a C^\infty ... 0answers 18 views ### piecewise linear system analysis with a switching boundary (plane) Let's say there's a piecewise linear system having a switching plane as shown in the figure(line in this example). And the initial condition of trajectory is given by a line segment X_0 at t=0. X_0... 0answers 123 views ### Uniform distribution modulo 1 and probability [closed] Define counting function A(E; N; \omega) as the number of terms x_n, 1\leq n\leq N, for which \{x_n\}\in E. Then the sequence \omega=(x_n), n=1,2,..., of real numbers is said to be uniformly ... 1answer 118 views ### 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 ... 0answers 262 views ### The busy Star Guardian On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of 1 in their ... 2answers 352 views ### About Lie group G has this escape property？ Every Lie group G has the following escape property: For every x \ne e in a sufficiently small neighborhood U of the identity e in G, there is a integer n such that x^n is not in U. ... 0answers 140 views ### Dynamical degree and spectral radius Let X be a smooth, projective surface over an algebraically closed field k of characteristic zero, and let f \in \mathrm{Bir}(X) a birational map. Let's denote f_{\ast} : \mathrm{NS}(X) \... 0answers 73 views ### Homoclinically related hyperbolic periodic points gives the same pesin homoclinic class up to null sets In MINIMALITY AND STABLE BERNOULLINESS IN DIMENSION 3 by Nunez and Hertz, the first paragraph in the proof of Corollary 2.4 says the above statement follows by using a "\lambda-lemma". ... 0answers 87 views ### Selecting a suitable Lyapunov function for the following systems? i) SI MODEL Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where N=S+I is the total population. If ... 1answer 547 views ### Searching for the proof of a certain claim in Arnold's ODE book from 1992 I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition). On pages 12-13 he writes the following: Consider the following ordinary ... 0answers 84 views ### Sequences generated from commuted quaternions and general commuted linear transformations Given a pair of non-commuting linear transformations, A and B, define the "next pair" in a sequence as A*B and B*A. I am interested in finite cycles (i.e., the sequence eventually ... 3answers 371 views ### Count of non-trivial ergodic measures of a topological dynamical system Given a compact Hausdorff space X and a continuous mapping \varphi: X \to X. We denote by C(X) the space of continuous functions f: X \to \mathbb{C}. A probability measure \mu on the Borel-\... 0answers 50 views ### Holomorphic dynamical systems defined on a contractible bounded open subset of \Bbb{C}^n Let U be a contractible bounded open subset of \Bbb{C}. There is a standard classification of possible dynamical behaviors of holomorphic maps f:U\rightarrow U: Attracting Case: There is an ... 1answer 141 views ### Equivalent definitions of strongly proximal action Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar, Kennedy and Ozawa: I have two questions: (1) What ... 1answer 199 views ### Does an “almost weakly mixing” transformation admit a non-null ergodic component? Problem set up: Let \mathbf X := (X, \mathcal A, \mu) be a standard probability space. We say that a measure preserving transformation T on \mathbf X is \varepsilon-almost weakly mixing if for ... 1answer 177 views ### Irrational rotations are rank 2 by intervals without spacers Let \alpha be an irrational number, and R_\alpha be the rotation by \alpha, that is R_\alpha(x)=x+\alpha\bmod 1. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ... 0answers 59 views ### Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds Let \tilde{M} be the universal cover of a pinched\ negatively curved manifold M and \Gamma=\pi_{1}(M) its fundamental group and \partial \Gamma =\partial \tilde{M} its Gromov boundary. When M... 0answers 51 views ### Solve (A(x).\nabla)u+cu=0 ِDoes the equation y\partial_x u(x,y)-x\partial_y u(x,y)+cu=0 have complex-valued compact-supported or vanishing-at-infinity C^1 solution defined on the whole plane without any singularity? Here ... 0answers 95 views ### Fell's absorption principle proof (for reduced crossed products) Consider the following proof from the book 'C^*-algebras and finite-dimensional approximations' by Brown-Ozawa. Let \Gamma be a discrete group and (A, \Gamma, \alpha) be a C^*-dynamical system.... 1answer 350 views ### How to analytically prove chaos Consider the following map \begin{align*} T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\ (x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\... 2answers 1k views ### Renormalization in physics vs. dynamical systems I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum ... 0answers 384 views ### Dynamics of a curious bijection of \mathbb N The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows: Given an ... 0answers 150 views ### Model theory and dynamical system (open problems) I am curious about the open problems which are between model theory and dynamical system. I mean the open problems that are interesting for both groups and there are some evidences showing there might ... 0answers 41 views ### Self-maps (dynamical systems) in several variables induced by functions X^{n+1}\to X Self-maps F:X\to X can be viewed as dynamic systems. A function f:X^{n+1}\to X induces a self-map F:X^{n+1}\to X^{n+1}, F(x_0\dots x_n):= (x_1, \ldots, x_n, f(x)) $$for every x:=(x_0, \... 2answers 279 views ### Textbooks or lecture notes about mean field games I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ... 0answers 108 views ### inverse of moment-generating function in terms of moments Let \{h_i\} be decreasing sequence of n positive reals. Define distribution p(X=h_i)\propto h_i and let g(s)=E_X[e^{sX}] be the moment generating function. For instance, for h=\{1,\frac{1}{4},... 0answers 176 views ### Measure concentrated on the \omega-limit set Let (X,F) be a dynamical system with X a compact metric space and F: X\to X continuous. By \omega-limit set of a subset A\subset X I mean:$$\omega(A):= \bigcap_{n=0}^\infty \left(\overline{\...
Let $(M,\omega,H)$ be a Hamiltonian system and assume that $\gamma$ is a periodic orbit on a regular energy hypersurface. Then the regular orbit cylinder theorem (see for example Abraham/Marsden: ...
We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...