Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations.

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3
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1answer
118 views

The relations between conservative part and conservativity

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ...
6
votes
2answers
267 views

Iterates converging to a continuous map

I have no doubt that the following observation is quite well known. Let $\varphi:[0,1]\to [0,1]$ be a continuous map. Assume that the iterates $\varphi^n$ converge pointwise to some continuous map ...
1
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0answers
108 views

Adding a damping term to a dynamical system or Markov process: what happens to invariant measures?

Consider the continuous-time Markov process on ${\mathbb R}^n$ described by the SDE $\dot{x}(t) = F(x(t)) + \xi(t)$ where $F:{\mathbb R}^n \to {\mathbb R}^n$ is a smooth mapping, and $\xi(t)$ is a ...
4
votes
2answers
129 views

Chain Recurrent Set of a Isometry

Let be $T:X\to X$ a topological dynamical system, $X$ a compact space and $T$ is also a isometry. Let be $\mathcal{R}(T)$ the chain recurrent set of $T$. Theorem: $\mathcal{R}(T)=X$ There is a ...
1
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2answers
295 views

Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
2
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0answers
164 views

Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

Background Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if $V$ is closed, $\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and $V_+ \cap (-V_+) = ...
8
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0answers
273 views

Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...
15
votes
10answers
2k views

Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
1
vote
1answer
341 views

Partial linearization near a hyperbolic fixed point--Classical scattering

I am currently reading the famous article "Universal Properties of Maps on an Interval" by Collet, Eckmann and Lanford related to the Feigenbaum-Coullet-Tresser universality. I am in particular ...
5
votes
2answers
208 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
2
votes
2answers
226 views

Equivalence of two definitions of Lyapunov exponents

I saw in articles two different definitions for Lyapunov exponents of a discrete dynamical system. Let's consider a discrete dynamical system $$ x_{k+1}=f(x_{k}),\quad x_{k}\in\mathbb{R}^{n},\quad ...
3
votes
0answers
83 views

Calculation of Lyapunov exponents for infinite systems of differential equations

Can you give an example of a function $\varphi$ and sequences $\{b_{i}\}$ and $\{a_{ij}\}$ for which one can calculate Lyapunov exponents of such the infinite system of differential equations and ...
17
votes
1answer
492 views

Can the expansion of a large integer in all bases consist of almost all zeroes?

Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...
1
vote
1answer
343 views

Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...
4
votes
2answers
332 views

Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as $$ t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu ...
1
vote
3answers
183 views

Applied examples of (non)uniformly hyperbolic and/or ergodic systems

I try to give reference to completely applied examples of (non)uniformly hyperbolic and/or ergodic systems. With completely applied I don't mean an irrational rotation on the torus but from other ...
4
votes
1answer
259 views

Characterising ergodicity of continuous maps

Hello all. Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic. Does there necessarily exist a Borel ...
1
vote
0answers
119 views

Detecting Non-Transversality

Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad ...
1
vote
1answer
237 views

Derivative of a random process

Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t. I consider applying a (stochastic)derivative operation to the random process. What is the ...
2
votes
6answers
312 views

Applications of discrete-time dynamics

Hello, I am a graduate student in the field of discrete-time dynamics. I am wondering about applications of this field outside of mathematics. More precisely, I would like to know if there are "real ...
5
votes
2answers
229 views

pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...
3
votes
0answers
129 views

Perturbation of Morse functions at critical points leaving stable manifolds invariant

Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858). Is there a way ...
6
votes
0answers
216 views

Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry. Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...
1
vote
1answer
93 views

Reference Request: Structural Stability of Gradient Fields

I am asking for a reference that contains a proof of Theorem 4, which is on page 315 of the following text: Hirsch, Morris W., and Stephen Smale. Differential equations, dynamical systems, and ...
1
vote
1answer
209 views

Integration by parts wrt. a Morse function on its basin of attraction

Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$ $$ \forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...
8
votes
2answers
349 views

Birational Automorphisms and infinite divisibility

Suppose $X$ is some algebraic variety. It can be over $\mathbb{C}$, but it doesn't have to (but char $0$ preferred). Is it possible that the additive group $\mathbb{Q}$ acts on it birationally, ...
2
votes
0answers
179 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
1
vote
0answers
149 views

Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets. He proved in Theorem 8.8 that in each finite partition of ...
10
votes
2answers
542 views

The height of an orbit under rational self-maps

I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I ...
1
vote
0answers
187 views

Interesting examples of minimal action on torus

Edit 1:This is a cross post on MSE. See math.stackexchange.com/q/289595/12952 Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor minimal ...
0
votes
0answers
97 views

Degree of freedom restricted by inequalities

Motivational example Consider a polyhedral graph $G$. A realization of $G$ is given by a convex polyhedron which is - essentially - characterized by the angles between the edges emanating from each ...
5
votes
0answers
159 views

Fibre Mixing for Dynamical Systems

Hi all, I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...
5
votes
1answer
465 views

“Explicit” examples of Irrational numbers very well approximated by rationnal numbers

This question relates to this one and that one. Some background In the setting of discrete holomorphic dynamics (say, Julia sets) an irrational $\lambda$ is said to be well approximated by rational ...
3
votes
1answer
228 views

A theory of bifurcation of braids ?

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional ...
1
vote
2answers
123 views

Relation between entropy of one-parameter group and single elements of this group

My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ...
7
votes
2answers
437 views

Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory: Invariant Measures Poincaré recurrence Theorem Invariant Measure For Continuous Transformations The Ergodic Theorems and Applications Ergodic ...
0
votes
3answers
149 views

Conley index for isolated invariant sets with no exit points

Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory. For the sake of simplicity let's think of vector fields ...
10
votes
1answer
284 views

Does the $n$-gonal billiards conjecture follow from the $m$-gonal conjecture when $m>n$?

For $n\ge 3$, define the $n$-gonal billiards conjecture as the statement All convex $n$-gons admit periodic billiard trajectories. To the best of my knowledge this question remains open for all ...
1
vote
1answer
258 views

Generalized Eigenvector in Dynamical System in Infinite Dimensions

Consider a system of linear delay differential equations: $$ \dot{z_1}(t) = z_1(t) + z_2(t-1) $$ $$ \dot{z_2}(t) = z_2(t) + z_3(t-1) $$ $$ \dot{z_3}(t) = z_3(t) - z_1(t-1) $$ The ...
2
votes
1answer
64 views

(A)periodicity and (In)dependence on the boundary condition for optimization problem related to ODE

The question is pair to MO117505 and translates some problem on error-correction codes to similar problem about differential operators. (See also If “force” is periodic does it imply “velocity” is ...
1
vote
1answer
106 views

(A)periodicity and (In)dependence on the boundary condition for some discrete analog of ODE (convolutional codes)

(See also MO117508, MO116611). This post describes somewhat real problem with convolutional codes. Let me first try to give brief and vague formulation of the question, later give details. Problem ...
1
vote
1answer
191 views

Terminology question in dynamical systems

Let $X$ be a topological space and let $f:X\rightarrow X$ be a continuous self-morphism of topological spaces. Let $Y$ be a closed $f$-stable subset of $X$, that is, suppose $f(Y)\subseteq Y$. ...
7
votes
2answers
350 views

Fixed objects of the M endofunctor on category Meas

Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces. As Gerald Edgar ...
11
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5answers
652 views

Arithmetic dynamics and dynamics on moduli spaces

The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces. In my dissertation, I have been ...
21
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3answers
4k views

Status of the 196 conjecture?

A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not ...
5
votes
2answers
396 views

sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
3
votes
0answers
229 views

A Dedekind Eta trajectory / horocyclic flow (Reference request)

I've been exploring the composition of essentially the Dedekind $\eta$-function with parabolic Möbius transformations, ...
5
votes
1answer
446 views

Numerical calculation of Arnold tongue

Hello. I am working on investigation of family of dynamical systems on the torus $$\dot{x}=\cos(x)+b\cos(t)+a$$ $$\dot{t}=1$$ and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$ I need to find ...
6
votes
0answers
177 views

Invariant curves of rational functions

Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function of degree at least 2 which maps $\gamma$ onto itself homeomorphically. The following examples of such ...
4
votes
2answers
263 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...