All Questions
Tagged with hyperbolic-dynamics hyperbolic-geometry
8 questions
5
votes
2
answers
309
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Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$
Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...
- 1,231
1
vote
0
answers
88
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Understanding logarithmic law for geodesics
I was reading this seminal paper
https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...
- 337
4
votes
1
answer
212
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When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
- 1,043
1
vote
1
answer
164
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Example of zero Lyapunov exponentes
Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function.
We say that an ...
- 1,043
5
votes
1
answer
988
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Ergodicity and mixing of geodesic and horocyclic flows
I am reading a theorem about the ergodicity and mixing of geodesic and horocyclic flows on unit tangent bundle of a compact hyperbolic surface. I find that there are two ways (be listed below) to ...
- 333
3
votes
1
answer
360
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Question on a proof of density of periodic orbits
In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...
user127549
4
votes
1
answer
246
views
Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
- 199
2
votes
0
answers
94
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Algebraic approach to prove the mixing property of Lorenz flow on hyperbolic surface
We knew that the noncompact subgroups of SL(2,$\mathbb{R}$) are mixing by Howe-Moore ergodicity theorem. I am curious about Lorenz flow, if we have a algebraic approach to prove the mixing property of ...
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