Continuous flows generate few homeomorphisms

2020 ◽  
Vol 63 (4) ◽  
pp. 971-983
Author(s):  
Wescley Bonomo ◽  
Paulo Varandas
Keyword(s):  
Topological Entropy ◽  
Continuous Flow ◽  
Compact Manifold ◽  
Necessary Conditions ◽  
Periodic Points ◽  
Compact Manifolds ◽  
Continuous Flows ◽  
Topological Obstructions ◽  
Volume Preserving ◽  
Dense Set

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

scholarly journals A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

2013 ◽  
Vol 34 (5) ◽  
pp. 1503-1524 ◽  
Author(s):  
THIAGO CATALAN ◽  
ALI TAHZIBI
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Topological Entropy ◽  
Upper Semicontinuity ◽  
Periodic Points ◽  
Surface Diffeomorphisms ◽  
Symplectic Diffeomorphisms ◽  
Symbolic Extension ◽  
Symplectic Diffeomorphism ◽  
Volume Preserving

AbstractWe prove that a${C}^{1} $generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that${C}^{1} $generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the${C}^{1} $topology.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

scholarly journals Topological entropy and partially hyperbolic diffeomorphisms

2008 ◽  
Vol 28 (3) ◽  
pp. 843-862 ◽  
Author(s):  
YONGXIA HUA ◽  
RADU SAGHIN ◽  
ZHIHONG XIA
Keyword(s):  
Topological Entropy ◽  
Topological Invariant ◽  
Compact Manifolds ◽  
Two Dimensional ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Lower Semi ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

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 Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

scholarly journals Hyperbolicity in the volume-preserving scenario

2012 ◽  
Vol 33 (6) ◽  
pp. 1644-1666 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
THIAGO CATALAN
Keyword(s):  
Analogous Result ◽  
Periodic Points ◽  
Axiom A ◽  
Volume Preserving

AbstractExtending a result of Mañé, Hayashi proved that every diffeomorphism$f$which has a$C^1$-neighborhood$\mathcal {U}$where all periodic points of any$g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

scholarly journals Chain recurrent points of a tree map

1999 ◽  
Vol 59 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Tao Li ◽  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Limit Set ◽  
Recurrent Point ◽  
Continuous Map ◽  
Tree Map

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if thew-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

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