New answers tagged ds.dynamical-systems
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Types of triangles admitting periodic billiard orbits
This paper in 2018 showed that all obtuse triangles with angles at most 112.3 degrees have periodic trajectories. It's also a computed assisted proof but not the same method as Richard Schwarz's. ...
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Does Bernoulli imply exponential mixing?
This is something that has confused me too, but let me try to answer anyway.
This question itself is a bit tricky to interpret. The reason is that "exponential mixing" really depends on the ...
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A mutation of the Collatz disease
commen: this is a copy of my MSE-answer which I linked to by the other comment. Because of the existent discussion in the comments here I thought today, it would be convenient to have it explicitely ...
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Central limit theorem for irrational rotations
I am surprised by the question. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$,
$$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big|
= \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \...
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Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Let's restrict attention to the case where the components of $S - \Lambda$ are all discs (with three or more cusps). (In particular, I assume that $\Lambda$ has no leaves which are simple closed ...
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Does every proximal dynamical system have zero topological entropy?
Maybe there is an easier example, but here is an example of a proximal system with positive entropy. The dynamical system is a so-called subshift $(X, \sigma)$, where $X$ is a closed shift-invariant ...
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Roadmap to Ergodic Theory
Concerning "landmark papers", Cosma Shalizi has documented their own pathway into ergodic theory, with an eye towards applications in statistical learning theory (but with many side branches)...
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Ergodic theory applied to number theory
The best example in my opinion is the work conducted mainly by Mariusz Lemańczyk, on problem related to the prime $k-$tuple conjecture. In the paper called $\mathscr{B}$-free sets and dynamics, prof. ...
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Ergodic theory applied to number theory
Terence Tao has a nice blog post about looking at the collatz conjecture from the viewpoint of Ergodic Theory
There are links in the post to other papers which also might be of interest to you.
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Ergodic theory applied to number theory
The interplay between ergodic theory and Number Theory owes a lot to the Abel Prize winner Hillel Furstenberg. So, I must suggest his book, Recurrence in Ergodic Theory and Combinatorial Number Theory....
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Help in understanding the singular system of linear forms and non escape of mass
Okay, these are just basic technical tidbits. Unsure if anything here is of research level.
$SO(n)$ acts transitively on the sphere of $\mathbb{R}^{n}$ (there's a slight ambiguity about $0$, but I ...
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Ergodic theory applied to number theory
For inspiration, you might enjoy reading The remarkable effectiveness of ergodic theory in number theory (2009).
The focus is on the use of the ergodic theory of homogenous flows to compute the ...
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Rate of convergence for Markov chain in random environment
Let us first observe what will happen in the simplest case when the random variables $A_\omega,\dots,A_{\sigma^n\omega}$ are independent and uniformly distributed on a finite set of matrices (the ...
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When does uniqueness of a stable equilibrium imply it is globally stable?
You can construct a smooth counterexample in $\mathbb{R}^2$: a function $f$ whose gradient flow has a unique equilibrium, which is also stable, but whose basin of attraction is not the entire plane. ...
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