**0**

votes

**0**answers

183 views

### Can we find an upper bound?

Let $f\in C^1(\mathbb R)$ with $f(0)=0$ and $|f'(x)|\le m$, where $m\in (1,2]$.
Let $x(0)\in\mathbb R$ be arbitrary, and define $x(n),y(n)$ recursively by
$$
x(n+1)=f(x(n)) , \quad\quad
...

**11**

votes

**1**answer

228 views

### Unusual digit sets that allow finite expansions for all (positive and negative) integers

Informal introduction
(If you don't like informal introductions, please skip to 'Mathematical formulation')
Whenever our 'decimal positional system' for writing numbers comes up in conversation, ...

**4**

votes

**1**answer

383 views

### Algebraic dynamics in finite fields

What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field ...

**3**

votes

**1**answer

117 views

### Packing measure and Kleinian groups

There has been "some" debate on the notion of fractal (as an illustration, see for example the discussion in this link). One of the possible notions includes relating Hausdorff dimension and packing ...

**2**

votes

**2**answers

250 views

### How to prove Liouville measure is invariant under geodesic flow?

Let $M$ be a complete n dimensional Riemannian manifold. $vol$ denotes the n dimensional Hausdorff measure. Let
$$
SM=\{(x,v)|x\in M, v\in T_xM, \|v\|=1\}
$$
be the unit tangent bundle of $M$. Then ...

**2**

votes

**0**answers

69 views

### Distal actions on coset spaces

Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point ...

**5**

votes

**0**answers

67 views

### Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims

I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...

**3**

votes

**0**answers

65 views

### Almost sure convergence of double nonconventional ergodic averages with respect to $L^p$ function

A famous result of J. Bourgain says that for a probability measure preserving system $(X,\beta,\mu,T)$, with $T_1$ and $T_2$ powers of $T$, we have that for $f_1$, $f_2\in ...

**0**

votes

**0**answers

39 views

### An almost periodic point must be a unifomly recurrent point?

$(X,G)$ is a topological semi-group action, $G$ is a topological (abelian) semigroup, and $X$ is a Hausdorff space.
$x\in X$ is called almost periodic of $(X,G)$, if for any neibourhood $U$ of $x$, ...

**0**

votes

**0**answers

79 views

### A question from One Dimensional Dynamics book by De-Melo and van-Strien

In One Dimensional Dynamics, on page 27 I don't understand how does $(1.7)$ follow; anyone care to explain this to me?
Thanks in advance.
I am adding some information from the text below:
We are ...

**7**

votes

**2**answers

152 views

### Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...

**7**

votes

**2**answers

124 views

### List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...

**1**

vote

**0**answers

53 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

**2**

votes

**0**answers

147 views

### Shape-related vector fields

Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$.
(of course it is not an equivalent relation).
...

**3**

votes

**1**answer

71 views

### Convergence of local stable manifolds

This question is a kind of local version of a previous post (MO224171).
Let $\ f_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, converging, in the $C^\infty$-topology, to a limit Morse ...

**6**

votes

**1**answer

104 views

### Stable manifolds of a sequence of Morse functions

Let $\ f_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, adequately converging (in the $C^2$-topology, say) to a limit Morse function $\ f$:
$$ f_n \to f \ .$$
At any critical point $\ p\ ...

**3**

votes

**1**answer

109 views

### The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$

Motivated by the following RG question we ask a related question as follows:
We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes ...

**27**

votes

**3**answers

812 views

### A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...

**5**

votes

**3**answers

342 views

### Ergodic theory: from Dynamics to Gibbs measure

I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.
I have a compact space $X$, a probability ...

**0**

votes

**0**answers

43 views

### Integrability of the orthogonal complement of a holomorphic vector field on $\mathbb{C}^{2}$

Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two ...

**0**

votes

**0**answers

158 views

### On a certain set of probability measures on a shift

Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where ...

**9**

votes

**1**answer

262 views

### The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...

**1**

vote

**1**answer

64 views

### Holomorphic vector field with infinite separatrix

Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...

**2**

votes

**2**answers

100 views

### “Dynamical” spectral gap for the orignal system out of the spectral gap for the induced system

I would like to prove presence of a spectral gap for the transfer (Ruelle-Perron-Frobenius) operator for some non-uniformly hyperbolic dynamical system on the unit interval. Suppose that I know how to ...

**1**

vote

**3**answers

205 views

### reference on complex dynamics

Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian ...

**6**

votes

**0**answers

122 views

### How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?

Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the ...

**5**

votes

**2**answers

225 views

### Noninvariance for a specific nonlinear oscillator

Consider the nonlinear system
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits ...

**0**

votes

**0**answers

85 views

### when the composition of two ergodic maps is ergodic?

I would like to know if there are sufficient criteria for the composition of two ergodic maps to be still ergodic.
My context is piecewise affine transformations of the torus in arbitrary dimensions

**20**

votes

**2**answers

647 views

### Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...

**0**

votes

**0**answers

74 views

### A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta ...

**16**

votes

**1**answer

666 views

### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := ...

**9**

votes

**1**answer

221 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...

**5**

votes

**0**answers

74 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**5**

votes

**0**answers

95 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

**5**

votes

**0**answers

61 views

### Is there a name for a bifurcation where a stable fixed point bifurcates into three fixed points: stable, unstable and saddle?

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by
$$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & ...

**4**

votes

**2**answers

256 views

### Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = ...

**2**

votes

**0**answers

121 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**2**

votes

**0**answers

87 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

**1**

vote

**0**answers

219 views

### A Perron-Frobenius problem

Let $A$ be an irreducible nonnegative matrix with column sums equal to 1.
Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...

**2**

votes

**1**answer

95 views

### Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems

I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, ...

**5**

votes

**0**answers

106 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**1**

vote

**1**answer

84 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

**6**

votes

**1**answer

307 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**4**

votes

**1**answer

63 views

### Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...

**3**

votes

**0**answers

154 views

### Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...

**7**

votes

**0**answers

124 views

### Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...

**1**

vote

**0**answers

54 views

### Ordering periodic orbits

I want to prove the proposition:
Proposition- Let $f:I \to I$ be continuos, and let f have a (2n+1)- periodic orbit {$x_{k}=f^{k}(x_{0})$, $k=0,1,\dots,2n$}, but no (2m+1)-periodic orbit for ...

**2**

votes

**1**answer

122 views

### Multi dimensional symbolic dynamics

I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?

**1**

vote

**0**answers

46 views

### Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...

**4**

votes

**1**answer

212 views

### The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.
Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...