# Tagged Questions

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

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### 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 $\mathbb{R}^n$, ...
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### 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 ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
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### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
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### Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
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### Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$

My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean ...
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### Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary. Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary). ...
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### Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)? It seems that the second part of the Hilbert 16th problem is solved or is going to ...
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### Growth of the size of iterated polynomials

I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of ...
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### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by \begin{eqnarray*} s_0 &=& a, \\ s_1 &=& b, \text{and} \\ s_{n+2} &=& s_{n+1} ...