Tagged Questions

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\tau: ... 2answers 166 views Hyperbolic sets that are not locally maximal I would like, if possible, a simple example of a hyperbolic set that is not locally maximal. What kind of dynamic phenomenon should occur for the appearance of hyperbolic set that is not maximal. 1answer 174 views Hyperbolic sets I recently started reading about hyperbolic dynamics in the notes of L. Wen, http://www6.cityu.edu.hk/rcms/publications/ln5.pdf and in this (page 8) there is the following statement: If the ... 2answers 400 views Variational Principle for the Entropy Theorem: Let be$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 ... 1answer 371 views Conley Theorem (or fundamental theorem of dynamical systems) Notations:$\mathcal{R}(f)$denotes the chain recurrent set of$fNW(f)$denotes the non wandering set of$fR(f)$denotes the recurrent set of$f$($x: x\in \omega(x)$) Given compact ... 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 ... 2answers 279 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 ... 2answers 135 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 ... 1answer 349 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 ... 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 ... 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 ... 2answers 463 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 ... 0answers 120 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 ... 0answers 169 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 ... 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 ... 1answer 279 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 ...
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(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 ...
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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 ...
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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 $g$ ...
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$\mathbb{S}^2$ 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 $[0,1)$, ...
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 ...