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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.
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
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 …
10
votes
Accepted
References on Lie groups and dynamical systems
The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours:
Smooth dynamics, like others have stated Hamiltonian dyanmics and differential equations.
Applicati …
8
votes
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem.
Actually if you will investigate Ratner's original formula …
7
votes
rate of equidistribution of the horocycle flow for $SL(2, \mathbb{Z})$
While Peter Humphries' answer is entirely correct for the question asked by the OP, the technique indicated there is far from addressing the most general situation.
The most basic technique towards t …
- 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
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 …
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
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
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
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
2
votes
How to show the geodesic orbit of a badly approximable number are/are not homogeneously equi...
A number is in BA if its orbit is bounded. Any such orbit closure must contain a full $A=\langle g_t\rangle$ orbit. By examining the possible subgroups, any such hypothetical $H$, as a stability group …
- 2,459
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
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