16
votes
2answers
535 views

Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...
1
vote
0answers
187 views

The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
0
votes
1answer
91 views

Non-convergence of ergodic measures with positive entropy

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let ...
1
vote
1answer
72 views

Example of non-convergence of iteration of measures

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes ...
1
vote
1answer
89 views

Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities". Let $f\colon \mathbb{C}^n\to ...
4
votes
1answer
125 views

Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...
6
votes
6answers
537 views

Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure. I wanted to have a ...
1
vote
0answers
78 views

Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system $$ \left\{ \begin{array}{l} \dot{x}=f(x), \\ x(t_0)=x_0, ...
3
votes
0answers
106 views

Nonexistence of Limit Cycle

Consider a planar dynamical system described in polar coordinates as $$ \left\{ \begin{array}{ll} \dot{\theta}=\Delta - r \sin \theta,\\ \dot{r} = - r + 1 + \cos \theta, \end{array} \right. $$ where ...
7
votes
1answer
353 views

Raphael Douady's thesis: Applications du théorème des tores invariants

Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts. According Wikipedia, he proves of the equivalence of KAM ...
6
votes
1answer
188 views

Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
3
votes
0answers
69 views

Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...
3
votes
1answer
160 views

Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
2
votes
1answer
114 views

Does conjugacy preserve the set of synchronizing blocks?

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then ...
2
votes
2answers
139 views

Mixing coded systems and period of their graph presentations

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8. ...
3
votes
2answers
135 views

Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce properties analogues to those of i.i.d. random variables. Call $X\doteq [0,1].$ Examples of decay of correlation ...
2
votes
0answers
98 views

Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups. (see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full) Is there a similar ...
2
votes
1answer
184 views

Uniform convergence of Birkhoff averages and unique ergodicity

I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET): Let $T$ be a ...
1
vote
1answer
171 views

Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations $$ \sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n, $$ has ...
2
votes
1answer
299 views

Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states $$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$ where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...
3
votes
0answers
173 views

The $\Omega$-Stability Theorem

I'm currently studying the $\Omega$-Stability Theorem: Theorem: If $\mathcal{R}(f)$ has a hyperbolic structure then $f$ is $\mathcal{R}$-stable. Some explanations about the statement: $f$ is a $C^1$ ...
1
vote
0answers
53 views

On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by $$\left\{ \begin{array}{l} \dot{\theta}_1 = \omega_1 - ...
2
votes
0answers
119 views

Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...
6
votes
3answers
287 views

Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof (or a reference to a proof) of the following theorem: Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure ...
3
votes
1answer
152 views

Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...
1
vote
0answers
83 views

Volume Function on Banach Spaces

I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant. Let $X$ be a Banach space with ...
6
votes
1answer
291 views

What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...
3
votes
1answer
194 views

Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems: Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...
1
vote
1answer
195 views

Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)

I have difficulties finding an appropriate reference for the following question (which I hope that it to be true). Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: ...
3
votes
2answers
161 views

Hyperbolic sets that are not locally maximal

I would like, if possible, a simple example of a hyperbolic set that is not locally maximal. What kind of dynamic phenomenon should occur for the appearance of hyperbolic set that is not maximal.
3
votes
1answer
173 views

Hyperbolic sets

I recently started reading about hyperbolic dynamics in the notes of L. Wen, http://www6.cityu.edu.hk/rcms/publications/ln5.pdf and in this (page 8) there is the following statement: If the ...
6
votes
2answers
395 views

Variational Principle for the Entropy

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$ Question: The above theorem is the famous variational principle ...
5
votes
1answer
357 views

Conley Theorem (or fundamental theorem of dynamical systems)

Notations: $\mathcal{R}(f)$ denotes the chain recurrent set of $f$ $NW(f)$ denotes the non wandering set of $f$ $R(f)$ denotes the recurrent set of $f$ ($x: x\in \omega(x)$) Given compact ...
3
votes
0answers
59 views

Do identical orbit tiles imply identical combinatorial types?

Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...
6
votes
2answers
276 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 ...
4
votes
2answers
134 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
vote
1answer
348 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 ...
1
vote
0answers
121 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
97 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 ...
7
votes
2answers
455 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 ...
1
vote
0answers
118 views

Rigid-body in a central field: orbital and attitude motion

Question I would like to find a nice set of explicit coordinates for the family (parametrised by angular momentum) of reduced systems representing a rigid-body in a central field in which the orbital ...
1
vote
0answers
164 views

How Markus–Yamabe implies Jacobian ?

To make myself precise, I would like to recall some backgrounds. (Markus-Yamabe, $\mathrm{MY}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$ with $f(0)=0$ and $Df$ everywhere Hurwitz stable (the ...
9
votes
3answers
1k views

Is there a categorical treatment of dynamical systems?

Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$? More precisely, is there a category whose ...
3
votes
1answer
273 views

Strata of quadratic differentials from rational billiards

Given a quadratic differential $q$ on a surface of genus $g$, we say that $q\in \mathcal Q(k_1,\ldots,k_n)$ if $q$ has $n$ distinct zeroes of order $k_1,\ldots,k_n$ respectively. The set $\mathcal ...
26
votes
1answer
968 views

solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.) I saw this amusing derivation ...
2
votes
2answers
486 views

Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood. I am considering a variant, which I call midpoint lattice polygons. Start with a sequence of ...
1
vote
0answers
259 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
2
votes
2answers
159 views

$\mathbb{S}^2$ equivalent to frac$(n \alpha)$ equidistribution on $\mathbb{S}^1$

Let $\operatorname{frac}(x) = x - \lfloor x \rfloor$ be the fractional part of $x$. Then, for $\alpha$ irrational, $\operatorname{frac}(n \alpha)$, $n=1,2,\ldots$, distributes randomly in $[0,1)$, ...
3
votes
3answers
274 views

Cryptography and iterations

Hi, Here is a question in cryptography which is probably naive, and a reference request. I was wondering about the following key-exchange scheme, which is a variant on Diffie-Hellman. Consider a ...
20
votes
4answers
878 views

Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle ...