All Questions
Tagged with gr.group-theory ds.dynamical-systems
79 questions
32
votes
3
answers
3k
views
Is there a reset sequence?
There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...
- 25.1k
30
votes
2
answers
2k
views
Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$
Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ \...
- 19.6k
22
votes
13
answers
7k
views
Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)?
Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...
- 649
18
votes
1
answer
2k
views
Rokhlin lemma for arbitrary infinite groups.
Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action ...
- 3,897
16
votes
3
answers
1k
views
What are some interesting examples of non-classical dynamical systems? (Group action other than $\mathbb{Z}$ or $\mathbb{R}$ )
By classical dynamical system, I mean a measure space together with a measurable action of the integers or the reals. Of course, this action is often interpreted as evolution with respect to discrete ...
16
votes
1
answer
502
views
Group actions and "transfinite dynamics"
$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
- 4,265
15
votes
1
answer
1k
views
In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?
This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
15
votes
0
answers
477
views
Diffeomorphisms of $\mathbf R^n$
Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.
Question: Is there an example to ...
- 1,782
14
votes
1
answer
956
views
On the iterated automorphism groups of the cyclic groups
Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
13
votes
3
answers
882
views
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
12
votes
1
answer
553
views
Topological amenability vs amenability of an action
Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...
- 121
12
votes
0
answers
268
views
If two group actions lead to the same orbifold, are they conjugate?
In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...
- 616
11
votes
1
answer
925
views
About positive upper density
For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...
- 121
11
votes
0
answers
258
views
Minimal actions commuting with amenable actions of $\mathbb{F}_2$
For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
- 211
11
votes
0
answers
203
views
Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
- 15.5k
10
votes
0
answers
466
views
For a group with one end does the property of connected spheres follow?
One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group $G$ with generators $g_1, \ldots g_n$....
- 1,143
8
votes
3
answers
1k
views
amenable equivalence relation generated by an action of a non-amenable group
Question. Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ ...
- 3,897
8
votes
4
answers
987
views
Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$
The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace \infty\rbrace$....
- 11.2k
8
votes
2
answers
684
views
Birational Automorphisms and infinite divisibility
Suppose $X$ is some algebraic variety. It can be over $\mathbb{C}$, but it doesn't have to (but char $0$ preferred).
Is it possible that the additive group $\mathbb{Q}$ acts on it birationally, ...
- 273
8
votes
2
answers
404
views
Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations"
Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a ...
- 7,328
8
votes
1
answer
1k
views
Classification of countable subgroups of the circle
Is there a classification of all countable subgroups of the circle $\mathbb{T} \simeq \mathbb{R}/\mathbb{Z}$?
It seems that there are quite a lot of them, e.g.:
cyclic subgroups $\{a^n\colon n\in\...
- 170
8
votes
4
answers
339
views
Iteration cycles of Z_n weights in path graphs: Why cycles of length 182 for a 6-node path?
Assign to the $n$ nodes of a path graph vertex weights
forming a permutation of $(0,\ldots,n{-}1)$.
Now iterate the following update repeatedly:
Each node sums the weights of its neighbors, and that ...
- 151k
7
votes
1
answer
811
views
Actions orbit equivalent to profinite ones
Let $G$ be a countable discrete residually finite group.
Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?
Ozawa and Popa introduced the concept of ...
7
votes
2
answers
585
views
Rotation numbers for amenable group actions on the circle
Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + \mathbb{Z}...
- 616
7
votes
0
answers
286
views
Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?
Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
user35370
7
votes
0
answers
305
views
Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action
Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
- 2,560
6
votes
2
answers
2k
views
Another question about amenability and Følner sequences
Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$,
$$\...
- 2,452
6
votes
2
answers
379
views
About Lie group $G$ has this escape property？
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
- 71
6
votes
2
answers
411
views
pointwise ergodic theorem and mean sojourn time
Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic theorems. ...
- 3,897
6
votes
1
answer
224
views
Topologically mixing cellular automata on groups
For which group-alphabet pairs $(G, A)$ does $(G, A^G)$ admit a topologically mixing cellular automaton?
Definitions:
Let $G$ be a (discrete) group. An alphabet is a finite set of cardinality at ...
- 6,652
5
votes
1
answer
448
views
Automorphisms of $\pi_1$ induced by pseudo-Anosov maps
Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.
For any $\...
- 23.5k
5
votes
2
answers
228
views
Bound on the period of the identity (in a free group) for an automorphism followed by left-multiplication
Let $F$ be a finite-rank free group, $g$ an element of $F$ and $\Phi\colon F \to F$ an automorphism. Consider the dynamical system $\psi_g\colon F \to F$ defined by $x \mapsto g\Phi(x)$. Say that $g$ ...
- 1,996
5
votes
2
answers
297
views
Seeking to understand meaning of "von Neumann spectrum" in a paper of Bader–Furman–Shaker
In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page)
I find that towards the end of the ...
- 2,125
5
votes
1
answer
169
views
Furstenberg decomposition for non-compact spaces
Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal ...
- 4,728
5
votes
0
answers
203
views
Terminology question in group actions
Given a continuous group action $G \times X \rightarrow X$ on a topological space $X$, is there a standard term for the subsets $K \subset X$ for which
Every open neighborhood of $K$ intersects every ...
- 13.5k
5
votes
0
answers
210
views
Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
- 356
5
votes
0
answers
135
views
Possible homogeneity of infinite dimensional Sierpinski carpet analogues?
Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion.
Construct subsets $S_n$ by removing points from $H$ if for any $m$,
at least $n$ of the ...
- 17.6k
5
votes
0
answers
352
views
"topological" conjugacy of group automorphisms
In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...
- 3,897
4
votes
2
answers
363
views
Complexification or 'real'ization of Mapping Class group.
So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
- 327
4
votes
1
answer
1k
views
Are all Nilmanifolds quotients of Heisenberg Group
I've been reading some wonderful blog entries where Terry Tao and Ben Green prove some generalizations of Weyl Equidstribution using a "higher" Fourier Analysis. Unfortunately, all the information I ...
- 22.8k
4
votes
1
answer
314
views
Properties of the spectrum of the Koopman representation
Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an ...
- 111
4
votes
1
answer
139
views
Asymptotic colouring of edges and vertices, and untwisting cocycles
This question regards colourings on edges and vertices on countable directed multigraphs.
We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...
4
votes
1
answer
332
views
Kac's lemma for amenable group actions
The classical Kac's lemma says the following.
Let $(X,\mu)$ be a probability space and $T$ a measure preserving transformation. Assume $A\subset X$ has positive measure. Then $$\sum_{k\ge 1} k\mu(A_k)...
- 41
4
votes
0
answers
108
views
Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
- 1,075
4
votes
0
answers
327
views
Amenable groups acting on the real line, that are not subexponentially-amenable
In the literature, there are several examples of solvable groups acting faithfully by order-preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth with ...
3
votes
1
answer
145
views
Topological amenability of actions - forgetting topology
Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
- 536
3
votes
1
answer
237
views
Invariant measure of a subgroup
Let $G$ be an abelian group with a $G$-invariant metric $d$. Let $H$ be a countable dense subgroup of $G$. Let $\mu$ be a non-atomic $\sigma$-finite Borel measure on $G$ that is $H$-invariant. Must it ...
- 59
3
votes
3
answers
569
views
Free ergodic probability measure-preserving actions of the free group
Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.
An action of $\Gamma$ on $X$ is:
essentially free if for all $g \in \Gamma \setminus \{e \}$,...
3
votes
1
answer
361
views
Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
- 175
3
votes
1
answer
219
views
Action of homeomorphism on real line
An element $f ∈ Homeo^+(\mathbb{R})$ is said to be of :
(1) type A, if it has a trivial germ at ∞ and it does not fix any point in an
interval of the form (−∞, r).
(2) type B, if it has a trivial ...
- 1,755