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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:29:21 +0000Regularity of the Anosov Splitting and A New Description of the Margulis Measure
https://resolver.caltech.edu/CaltechETD:etd-05232007-105707
Authors: {'items': [{'id': 'Hasselblatt-Boris', 'name': {'family': 'Hasselblatt', 'given': 'Boris'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/qkw4-xf63
<p>The Anosov splitting into stable and unstable manifolds of hyperbolic dynamical systems has been known to be Holder continuous always and differentiable under bunching or dimensionality conditions. It has been known, by virtue of a single example, that it is not always differentiable. High smoothness implies some rigidity in several settings.</p>
<p>In this work we show that the right bunching conditions can guarantee regularity of the Anosov splitting up to being differentiable with derivative of Holder exponent arbitrarily close to one. On the other hand we show that the bunching condition used is optimal. Instead of providing isolated examples we prove genericity of the low-regularity situation in the absence of bunching. This is the first time a local construction of low-regularity examples is provided.</p>
<p>Based on this technique we indicate how horospheric foliations of nonconstantly curved symmetric spaces can be made to be nondifferentiable by a smoothly small perturbation.</p>
<p>In the last chapter the Hamenstadt-description of the Margulis measure is rendered for Anosov flows and with a simplified argument. The Margulis measure arises as a Hausdorff measure for a natural distance on (un)stable leaves that is adapted to the dynamics.</p>https://thesis.library.caltech.edu/id/eprint/1986Rigidity of three measure classes on the ideal boundary of mainifolds with negative curvature
https://resolver.caltech.edu/CaltechTHESIS:04112011-144739148
Authors: {'items': [{'name': {'family': 'Yue', 'given': 'Chengbo'}, 'show_email': 'NO'}]}
Year: 1991
DOI: 10.7907/4wkq-q530
On the ideal boundary, ∂M, of the universal covering of M of a negatively curved closed Riemannian manifold M, there exist three natural measure classes: the harmonic measure class {v_x}_(x∈M), the Lebesgue measure class {m_x}_(x∈M), the Bowen-Margulis measure class {u_x}_(x∈M).
A famous conjecture (by A. Katok, F. Ledrappier, D. Sullivan) states that the coincidence of any two of these three measure classes implies that M is locally symmetric. We prove a weaker version of Sullivan’s conjecture: the horospheres in M have constant mean curvature if and only if m_x=v_x for all x ∈ M.
In investigating these rigidity problems, we come across a class of integral formulas involving Laplacian Δ^u along the unstable foliation of the geodesic flow. One of which is ^∫_(SM) (Δ^u φ
+ < ∇^u log g, ∇^u φ >)dm = 0. Using these formulas, many rigidity problems are discussed, including (i) a simple proof of Hamenstädt’s lemma 5.3 which avoids her use of stochastic process, (ii) two functional descriptions of those manifolds which have horospheres with constant mean curvature: the horospheres in M have constant mean curvature if and only if ^∫_(SM) Δ^u φdm = 0 for all φ in C^2_u(SM) or ^∫_(SM) Δ^(su) φdm = 0 for all φ in C^2_(su)(SM).
Finally, we study ergodic properties of Anosov foliations and their applications to manifolds of negative curvature. We obtain an integral formula for topological entropy in terms of Ricci and scalar curvature. We also show that the function c(x) in Margulis’s asymptotic formula c(x) = lim_(R→∞ e^(-hR)S(x,R) is almost always nonconstant. In dimension 2, c(x) is a constant function if and only if the manifold has constant negative curvature. Generally, if the Ledrappier-Patterson-Sullivan measure is flip invariant, then c(x) is constant.
https://thesis.library.caltech.edu/id/eprint/6300Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov R^n Actions
https://resolver.caltech.edu/CaltechTHESIS:07122011-075820059
Authors: {'items': [{'id': 'Qian-N', 'name': {'family': 'Qian', 'given': 'Nantian'}, 'show_email': 'NO'}]}
Year: 1992
DOI: 10.7907/M7JX-ZV02
An action of a group Г on a manifold M is a homomorphism ρ from Г to Diff(M). ρo is locally rigid if the nearby homomorphism ρ, ρ(γ) = h o ρ0, (γ) 0h^(-1) for some h Є Diff(M) and for all, γ Є Г. In other words, ρ0 is isolated from other actions up to a smooth conjugation.
In this thesis we studied some standard group actions on a broader class of manifolds, the free, k-step nilmanifolds N(n, k); we obtained that the standard SL(n, Z) action on N(n,2) is locally rigid for n = 3, and n ≥ 5.
We recall that N(n,1) = T^n. Hence, our results are the generalization to the local rigidity result for the standard action on torus T^n.
We observed also, for the first time, that for discrete subgroups Aut(n, 2) of a Lie group, which is not even reductive, the action on N(n,2) is deformation-rigid
for n = 3, and n ≥ 5.
We also investigated the dynamics of Anosov R^n actions and obtained a number of results parallel to those of Anosov diffeomorphisms and flows. E.g., the strong stable (unstable) manifold for a regular element is dense iff the action is weakly mixing (for a volume-preserving action); an Anosov action with no dense, strong stable (unstable) manifold can always be reduced to suspension of the action
mentioned above; there are two compatible measures to the Anosov actions.
https://thesis.library.caltech.edu/id/eprint/6538Birkhoff periodic orbits, Aubry-Mather sets, minimal geodesics and Lyapunov exponents
https://resolver.caltech.edu/CaltechTHESIS:11122012-095207855
Authors: {'items': [{'id': 'Chen-W-F', 'name': {'family': 'Chen', 'given': 'Wei-Feng'}, 'show_email': 'NO'}]}
Year: 1993
DOI: 10.7907/g2ca-jx87
<p>Aubry-Mather theory proved the existence of invariant circles and invariant
Cantor set (the ghost circles) for the area-preserving, monotone twist maps of
annulus or of cylinders. We are interested in higher dimensional systems. The
celebrated KAM theorem established the existence of invariant tori for small perturbations
of integrable Hamiltonian systems with nondegenerate Hamiltonian
functions, but said nothing about the missing tori. Bernstein-Katok found the
Birkhoff periodic orbits, which are viewed as the traces of missing tori, for the
system in the KAM theorem but under the stronger condition that the Hamiltonian
function is convex. We find the "isolating block", a structure invented by
Conley and Zehnder, to demonstrate the existence of Birkhoff periodic orbits for
the KAM system.</p>
<p>In the second part, we wanted to establish the existence of minimal closed
geodesic which is hyperbolic on the surface of genus greater than one. There is
strong evidence that such geodesics exist. We find a curvature condition for the
minimal closed geodesic, thus furnishing further evidence.</p>https://thesis.library.caltech.edu/id/eprint/7264