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Results tagged with ds.dynamical-systems
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user 8857
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
3
votes
Accepted
Uniquely ergodicity and polynomial ergodic average
This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$-functions) for Kronecker systems simply by van-der-Corput trick.
In general, those averages converg …
- 2,459
6
votes
Examples of transformations that are totally ergodic but not weakly mixing?
Totally ergodic is equivalent to not having rational eigenvalues (I guess a suitable reference for this is Eli's book).
Hence basically the Kronecker factor of such a system will be "essentially" the …
- 2,459
1
vote
Accepted
An angle between two vectors in Oseledets theorem
Ok, let take $v$ in the bundle at $x$. We may decompose $v=v^{u}+v^{s}\in E^{u}\oplus E^{s}$.
Assume without loss of generality that $\lVert v\rVert=1$.
Applying $A^{i}$, using equivariance and Osceld …
- 2,459
0
votes
Accepted
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 gue …
- 2,459
5
votes
Book recommendation for ergodic theory and/or topological dynamics?
I second Siming Tu's recommendation for E-W book.
It is a well balanced book (regarding theory vs applications), it has nice appendix contains relevant theory from functional analysis, and it contains …
26
votes
Accepted
If there is a dense geodesic, are almost all geodesics equidistributed? Dense?
The first question is false as stated.
By Artin's encoding, geodesics on $SL_{2}(\mathbb{R})/SL_{2}(\mathbb{Z})$ corresponding to continued fractions, and the geodesic flow corresponds to the shift.
I …
- 2,459
1
vote
Some puzzles about the three conditions in a paper of D.Berend
For a start, I guess that one should be very familiar of the proofs of Furstenberg's diophantine result, as this paper generalizes this theorem.
Secondly, it might be of interest that Zhiren Wang in h …
- 2,459
1
vote
The identity element of a compact group is a limit point of any "polynomial sequence"
For getting the every-point statement, at-least in the compact abelian case (see Tao's comment above), one can either prove it by harmonic analytic approach (Weyl's equi. criterion + van der corput tr …
- 2,459
2
votes
Accepted
Basic question on minimal flows
Your question is about a theorem of Furstenberg.
About the definitions - obviously every periodic orbit is minimal, if exists, hence in the case the action is minimal, you won't have any periodic orb …
- 2,459
2
votes
Ratner's orbit closure for a unipotent semigroup
$\DeclareMathOperator\supp{supp}$The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided avera …
- 2,459
1
vote
Accepted
Compact group extension of a zero entropy system.
[I'm assuming G means K]
In case the elements of K are not measure preserving, I doubt that the question is correct.
In general, one can easily show that entropy decrease by moving into factors.
Her …
- 2,459
3
votes
Accepted
Furstenberg-Zimmer theorem: non-invertible systems
Posted as requested - consult the book by Manfred Einsiedler and Tom Ward - "Ergodic Theory with a view towards number theory" - published in GTM, especially in ch 7.
- 2,459
11
votes
Accepted
Furstenberg $\times 2 \times 3$ conjecture, bibliography
Well that will be some lengthy answer.
The first article that was published after the famous disjointness paper is another paper by Hillel called "Intersections of Cantor sets", it's related to the m …
1
vote
Accepted
Entropy equals zero?
EDIT - The answer below deals with an ergodic m.p.s
As this question got up-voted, I've decided to fuly write a solution, based on the sketch I've made in the comments.
Fix some $\varepsilon>0$ smal …
- 2,459
2
votes
Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension
Show that for a Bernoulli system, there exists ergodic (Bernoulli) measures of any given entropy (between 0 and full entropy). Pick such a measure with appropriate entropy as you would like. Recall t …
- 2,459