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Results tagged with ergodic-theory
Search options questions only
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user 6094
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
15
votes
2
answers
2k
views
Are rounded rectangle billiard dynamics ergodic?
Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.
(Image from Microwave_billiards_and_quantum_chaos.)
Q. Is it known that the billiard-ball dynami …
- 151k
1
vote
0
answers
65
views
"Saddle connection" on a translation surface
A saddle connection on a translation surface $\omega$ is a geodesic
in the flat metric determined by $\omega$ joining two zeros with no zeros in its interior.
Athreya, Jayadev S., and Howard Masur. …
- 151k
7
votes
2
answers
736
views
How quickly will billiard trajectories cluster?
Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a configur …
- 151k
20
votes
5
answers
1k
views
Iterated Circumcircle
Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no collin …
- 151k
28
votes
0
answers
822
views
Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by …
- 151k
18
votes
0
answers
480
views
Trapping lightrays with segment mirrors
Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors?
I posed this question in several forums before (e.g., here
and in an ea …
- 151k
17
votes
2
answers
2k
views
Random walk is to diffusion as self-avoiding random walk is to ...?
One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a …
- 151k
24
votes
2
answers
1k
views
Billiard dynamics for multiple balls
I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
coll …
- 151k
16
votes
3
answers
2k
views
A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $L_ …
- 151k
33
votes
4
answers
3k
views
Does there exist a shot in ideal pocket billiards?
Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard bal …
- 151k
5
votes
0
answers
163
views
Pocket billiards with balls in general position
There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does t …
- 151k