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

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        0answers
        11 views

        About the limit of transverse intersection

        Let $n$ be a fixed positive integer, and let $W^{s}(R_{q})$ and $W^{u}(R_{q})$ be the stable and unstable manifolds of a fixed point $R_{q}$ of a discrete 2-D mapping. Notice that the sequence $R_{q}$ ...
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        votes
        2answers
        197 views

        Newton method and Siegel disks

        I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?
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        votes
        1answer
        128 views

        Random reflections unexpectedly produce banded distributions

        Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
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        vote
        1answer
        120 views

        Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

        Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$. There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
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        votes
        1answer
        107 views

        Irreducible but not completely irreducible

        Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$). Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, ...
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        vote
        1answer
        130 views

        continuity entropy with respect gibbs measures

        Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
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        1answer
        25 views

        Marginal stability of discrete linear time-invariant system

        I have a question about marginal stability of a system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I would adapt the definition of marginal stability from this ...
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        57 views

        Stable commutator lengths of pseudo-Anosovs

        Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?
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        vote
        1answer
        66 views

        stationary measure for linear cocycle(random transformation matrices)

        Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...
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        0answers
        100 views

        How was the pair of pants introduced [closed]

        There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful? For example, does it have any application if we estimate the perimeter or ...
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        votes
        2answers
        277 views

        Minimal, uniquely ergodic but not Lebesgue-ergodic?

        So here's my question: Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is minimal uniquely ergodic with unique probability measure $\mu$ ...
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        votes
        1answer
        83 views

        Lyapunov exponent of singular values and operator norm

        Consider a product of i.i.d. $3\times 3$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$) acting on a non-zero vecor $V \in \mathbb{R^3}$, i.e. $$ A_{n}\cdots A_{1}...
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        2answers
        374 views

        Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

        Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice. I have heard that, for any real number $\...
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        0answers
        29 views

        definition of mixing component

        definition of ergodic component: consider stationary dynamical system $(X, \mathcal{B}, \mu, T)$, each ergodic component is $m(\cdot)=\mathbb{E}_{\mu}^{\mathcal{I}}\mathbf{1}_{(\cdot)}$, $\mathcal{I} $...
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        0answers
        85 views

        Is $\partial M_d$ continuously determined by $d$?

        This question is inspired by a question on math.stackexchange: https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089 The animation ...

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        山西福彩快乐十分钟
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