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Results tagged with ergodic-theory
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user 30691
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
7
votes
Accepted
Decay of cusps in geometrically finite groups
You can do it as an exercise, following the strategy in Dal'bo-Otal-Peigné "Série de Poincaré des groupes géométriquement finis", Israel J math. 118 (2000). See here, in french : http://www.lmpt.univ- …
- 987
3
votes
invariant measure of uniquely ergodic horocycle flow
The result of Marcus can be formulated in terms of strong (un)stable foliations of the geodesic flow in negative curvature.
The result, due to Bowen-Marcus in the compact case, to Roblin as soon as …
- 987
7
votes
Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
Another interpretation of this fact is as follows.
Consider the unit tangent bundle $PSL(2,\mathbb{Z})\backslash PSL(2,\mathbb{R})$ of the modular surface $PSL(2,\mathbb{Z}) \backslash \mathbb{H}^2$. …
- 987
0
votes
Approximating Subshifts From Below
I guess that you can find such a statement in the first articles of Omri Sarig about thermodynamical formalism.
If your shift is mixing, then the pressure/entropy of your shift coincides with the sup …
- 987
8
votes
Accepted
On the Birkhoff ergodic theorem for geodesic flows
The answer is NO. There is no chance for such a thing.
The reason is the following. The asymptotic behaviour of the ergodic average depends on the asymptotic geometric behaviour of the ... geodesic …
- 987
4
votes
Accepted
the union of local stable manifolds along local unstable manifolds
The local product structure says that for hyperbolic diffeos (it works also for flows), given two points $z$ and $z'$ in a small neighbourhood of $y$, then $W^s_\delta(z)$ and $W^u_\delta(z')$ interse …
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1
vote
Relation between entropy of one-parameter group and single elements of this group
A small precision :
If $\mu$ is an ergodic measure for the action of the one-parameter group
$H=(h_t)_{t\in\mathbb{R}}$, then for almost every $t\in \mathbb{ R}$, the single element $h_t$ is ergodi …
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3
votes
proofs of ergodicity of Sl(2, Z) action on R^2 without using duality
I recommand the work of Thomas Roblin "Ergodicité et équidistribution en courbure négative" Mémoires de la SMF 95 (2003), in addition to the references mentioned in the first answer.
In particula …
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2
votes
A follow up question related to entropy
In the case of the geodesic flow acting on the unit tangent bundle of a compact negatively curved manifold, if $a_n$ is the number of closed geodesics of length at most $n$, and $h$ the topological e …
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9
votes
What are the zero entropy invariant measures for an Anosov geodesic flow?
Another type of answer. It can be shown that generically (in the sense of Baire), an invariant probability measure for the geodesic flow is ergodic, of full support, with entropy zero, and not mixing. …
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