Anosov flows which are uniformly expanding at periodic points

1994 ◽  
Vol 14 (2) ◽  
pp. 299-304 ◽  
Author(s):  
Ursula Hamenstadt
Keyword(s):  
Compact Manifold ◽  
Periodic Points ◽  
Anosov Flow ◽  
Anosov Flows

AbstractA smooth transitive Anosov flow on a compact manifoldNwhich is uniformly (a, b)-expanding at periodic points for 1 <a<bis uniformly (a− ε,b+ ε)-expanding on all of N for all ε > 0.

scholarly journals Global cross sections for Anosov flows

2015 ◽  
Vol 36 (8) ◽  
pp. 2661-2674
Author(s):  
SLOBODAN N. SIMIĆ
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Anosov Flow ◽  
Anosov Flows ◽  
New Criterion ◽  
Volume Preserving

We provide a new criterion for the existence of a global cross section to a volume-preserving Anosov flow. The criterion is expressed in terms of expansion and contraction rates of the flow and generalizes known results of this type.

scholarly journals Approximate solutions of cohomological equations associated with some Anosov flows

1990 ◽  
Vol 10 (2) ◽  
pp. 367-379 ◽  
Author(s):  
Svetlana Katok
Keyword(s):  
Approximate Solutions ◽  
Geodesic Flows ◽  
Curved Surfaces ◽  
Anosov Flow ◽  
3 Dimensional ◽  
Negatively Curved ◽  
Anosov Flows ◽  
Higher Differentiability ◽  
Special Case ◽  
Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

scholarly journals Immediate conditional hyperbolicity in dynamical systems

1983 ◽  
Vol 3 (4) ◽  
pp. 627-647
Author(s):  
Joseph Rosenblatt ◽  
Richard Swanson
Keyword(s):  
Vector Bundle ◽  
Dynamical Systems ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Almost Periodic ◽  
Uniquely Ergodic

AbstractFor many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.

scholarly journals Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Jerzy Jezierski
Keyword(s):  
Lie Groups ◽  
Explicit Form ◽  
Homotopy Class ◽  
Unitary Group ◽  
Compact Manifold ◽  
Periodic Points ◽  
Necessary Condition ◽  
Minimal Realization ◽  
Compact Lie Groups

AbstractLet $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

Regularity at infinity of compact negatively curved manifolds

1994 ◽  
Vol 14 (3) ◽  
pp. 493-514
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Periodic Points ◽  
Geodesic Flows ◽  
Stable Foliation ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Measure Class

AbstractIt is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

PERIODIC POINTS OF ${\mathcal C}^1$ MAPS AND THE ASYMPTOTIC LEFSCHETZ NUMBER

1995 ◽  
Vol 05 (05) ◽  
pp. 1369-1373 ◽  
Author(s):  
JOHN GUASCHI ◽  
JAUME LLIBRE
Keyword(s):  
Fixed Point ◽  
Compact Manifold ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Periodic Points ◽  
Lefschetz Number

We study the set of periodic points and the set of periods of different classes of [Formula: see text] self maps of a compact manifold. We give sufficient conditions in order that these sets be infinite. Our main tools are the Lefschetz fixed point theory and homology.

Codimension one Anosov flows and a conjecture of Verjovsky

1997 ◽  
Vol 17 (5) ◽  
pp. 1211-1231 ◽  
Author(s):  
SLOBODAN SIMIĆ
Keyword(s):  
Cross Section ◽  
Compact Riemannian Manifold ◽  
Positive Answer ◽  
Stable Bundle ◽  
Anosov Flow ◽  
Hölder Continuous ◽  
Codimension One ◽  
Higher Dimensional ◽  
Anosov Flows ◽  
Volume Preserving

Let $ \Phi $ be a $C^2$ codimension one Anosov flow on a compact Riemannian manifold $M$ of dimension greater than three. Verjovsky conjectured that $ \Phi $ admits a global cross-section and we affirm this conjecture when $ \Phi $ is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of $\Phi$ is $ \theta $-Hölder continuous for all $ \theta < 1 $; (2) if the center stable bundle of $ \Phi $ is of class $ C^{1 + \theta} $ for all $ \theta < 1 $. We also show how certain transitive Anosov flows (those whose center stable bundle is $C^1$ and transversely orientable) can be ‘synchronized’, that is, reparametrized so that the strong unstable determinant of the time $t$ map (for all $t$) of the synchronized flow is identically equal to $ e^t $. Several applications of this method are given, including vanishing of the Godbillon–Vey class of the center stable foliation of a codimension one Anosov flow (when $ \dim M > 3 $ and that foliation is $ C^{1 + \theta} $ for all $ \theta < 1 $), and a positive answer to a higher-dimensional analog to Problem 10.4 posed by Hurder and Katok in [HK].

On the rigidity of quasiconformal Anosov flows

2007 ◽  
Vol 27 (6) ◽  
pp. 1773-1802 ◽  
Author(s):  
YONG FANG
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
Anosov Flow ◽  
Orbit Equivalent ◽  
Finite Covers ◽  
Hyperbolic Automorphism ◽  
Anosov Flows

AbstractWe develop further our study of quasiconformal Anosov flows in our previous (Y. Fang. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys.24 (2004), 1–23). For example, we prove the following result: Let φ be a transversely symplectic Anosov flow with dim  Ess≥2 and dim  Esu≥2. If φ is quasiconformal, then it is, up to finite covers, $C^\infty $ orbit equivalent either to the suspension of a symplectic hyperbolic automorphism of a torus or to the geodesic flow of a closed hyperbolic manifold.

Periodic orbits of transversal maps

1995 ◽  
Vol 118 (1) ◽  
pp. 161-181 ◽  
Author(s):  
J. Casasayas ◽  
J. Llibre ◽  
A. Nunes
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Zeta Function ◽  
Point Theorem ◽  
Compact Manifold ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Homology Groups ◽  
Lefschetz Numbers ◽  
Lefschetz Fixed Point Theorem

One of the most useful theorems for proving the existence of fixed points, or more generally, periodic points of a continuous self-map f of a compact manifold, is the Lefschetz fixed point theorem. When studying the periodic points of f it is convenient to use the Lefschetz zeta function Zf(t) of f, which is a generating function for the Lefschetz numbers of all iterates of f. The function Zf(t) is rational in t and can be computed from the homological invariants of f. See Section 2 for a precise definition. Thus there exists a relation, based on the Lefschetz fixed point theorem, between the periodic points of a self-map of a manifold f:M → M and the properties of the induced homomorphism f*i on the homology groups of M. This relation has been used in several papers, namely [F1], [F2], [F3] and [M]. In these papers, sufficient conditions are given for the existence of infinitely many periodic points in the case when all the zeros and poles of the associated Lefschetz zeta function are roots of unit. Here we restrict ourselves to maps defined on manifolds with a certain homology type. For transversal maps f defined on this class of manifolds, it is possible to extend the techniques introduced in [F1], [F3] and [M] in order to obtain information on the set of periods of f. We recover the above mentioned results of J. Franks and T. Matsuoka, and derive new results on the set of periods of f when the associated Lefschetz zeta function has zeros or poles outside the unit circle.

scholarly journals The number of periodic points of smooth maps

1989 ◽  
Vol 9 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Takashi Matsuoka
Keyword(s):  
Compact Manifold ◽  
Periodic Points ◽  
Sufficient Condition ◽  
Smooth Maps ◽  
Topological Condition

AbstractLetf: M → M be a C1 map on a compact manifold. We give a topological condition under which f has an even number of periodic points with a given period. We also obtain a sufficient condition, in terms of homology, for ƒ to have infinitely many periodic points.

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