scholarly journals Escape rates for special flows and their higher order asymptotics

2017 ◽  
Vol 39 (06) ◽  
pp. 1501-1530
Author(s):  
FABIAN DREHER ◽  
MARC KESSEBÖHMER
Keyword(s):  
Metric Spaces ◽  
Higher Order ◽  
Escape Rate ◽  
Convergence In Probability ◽  
Regularity Conditions ◽  
General Context ◽  
Scaling Invariance ◽  
Continuity Properties ◽  
Higher Order Asymptotics ◽  
Additional Regularity

In this paper escape rates and local escape rates for special flows are studied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfils certain scaling, invariance and continuity properties. In the context of metric spaces local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also, a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals Absence of high-energy spectral concentration for Dirac systems with divergent potentials

2005 ◽  
Vol 135 (4) ◽  
pp. 689-702 ◽  
Author(s):  
M. S. P. Eastham ◽  
K. M. Schmidt
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Regularity Conditions ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Local Maxima ◽  
Dirac Systems ◽  
Additional Regularity ◽  
Spectral Concentration

It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.

scholarly journals COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

2018 ◽  
Vol 98 (3) ◽  
pp. 422-433
Author(s):  
BORIS GOLDFARB ◽  
JONATHAN L. GROSSMAN
Keyword(s):  
Metric Spaces ◽  
General Context ◽  
Metric Geometry ◽  
Finitely Generated ◽  
Coherence Property ◽  
Finitely Generated Groups ◽  
Permanence Property ◽  
Permanence Properties ◽  
New Framework

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Finite powers of strong measure zero sets

1999 ◽  
Vol 64 (3) ◽  
pp. 1295-1306 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Measure Zero ◽  
General Context ◽  
Partition Relation ◽  
Compact Metric Space ◽  
Totally Bounded ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

scholarly journals Absence of high-energy spectral concentration for Dirac systems with divergent potentials

2005 ◽  
Vol 135 (4) ◽  
pp. 689-702
Author(s):  
M. S. P. Eastham ◽  
K. M. Schmidt
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Regularity Conditions ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Local Maxima ◽  
Dirac Systems ◽  
Additional Regularity ◽  
Spectral Concentration

It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.

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