scholarly journals Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 38
Author(s):  
Kazuhiro Sakai ◽  
Naoya Sumi
Keyword(s):  
Lebesgue Measure ◽  
Shadowing Property ◽  
Axiom A ◽  
Open Set ◽  
Cycle Condition

In this paper, it is proved that every diffeomorphism possessing the filtrated pseudo-orbit shadowing property admits an approximately shadowable Lebesgue measure. Furthermore, the C1-interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. As a corollary, it is proved that there exists a C1-open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

scholarly journals On approximating Lebesgue integrals by Riemann sums

1991 ◽  
Vol 33 (2) ◽  
pp. 129-134
Author(s):  
Szilárd GY. Révész ◽  
Imre Z. Ruzsa
Keyword(s):  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Function ◽  
Riemann Sums ◽  
Function Classes ◽  
Open Set ◽  
Measurable Set ◽  
Image Position ◽  
Good Sequences

If f is a real function, periodic with period 1, we defineIn the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ ℝ is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of ℒp is ℒp [0, 1]. For all real xwe haveif f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ ℒ1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ ℒp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) towhere S ⊂ ℕ is some “good” increasing subsequence of ℕ. Naturally, for different function classes ℱ ⊂ ℒ1 we get different meanings of being good. That is, we introduce the class of ℱ-good sequences as

Nonsingular star flows satisfy Axiom A and the no-cycle condition

2005 ◽  
Vol 164 (2) ◽  
pp. 279-315 ◽  
Author(s):  
Shaobo Gan ◽  
Lan Wen
Keyword(s):  
Axiom A ◽  
Cycle Condition

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

scholarly journals On the Gauss-Green theorem

1968 ◽  
Vol 8 (3) ◽  
pp. 385-396 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Lebesgue Measure ◽  
Inner Product ◽  
Open Set ◽  
The Individual ◽  
Image Position ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Dimensional Lebesgue Measure ◽  
Lebesgue Integration ◽  
Riemann Integration

In a previous paper [1], Green's theorem for line integrals in the plane was proved, for Riemann integration, assuming the integrability of Qx−Py, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx and Py. In the present paper, this result is extended to a proof of the Gauss-Green theorem for p-space (p ≥ 2), for Lebesgue integration, under analogous hypotheses. The theorem is proved in the form where Ω is a bounded open set in Rp (p-space), with boundary Ω; g(x) =(g(x1)…, g(xp)) is a p-vector valued function of x = (x1,…,xp), continuous in the closure of Ω; μv,(x) is p-dimensional Lebesgue measure; v(x) = (v1(x),…, vp(x)) and Φ(x) are suitably defined unit exterior normal and surface area on the ‘surface’ ∂Ω and g(x) · v(x) denotes inner product of p-vectors.

scholarly journals SRB measures for almost Axiom A diffeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2015-2043 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
RENAUD LEPLAIDEUR
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Compact Manifold ◽  
Invariant Set ◽  
Singular Set ◽  
Axiom A ◽  
Srb Measures ◽  
Hyperbolic Points

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

scholarly journals On two functionals involving the maximum of the torsion function

2018 ◽  
Vol 24 (4) ◽  
pp. 1585-1604 ◽  
Author(s):  
Antoine Henrot ◽  
Ilaria Lucardesi ◽  
Gérard Philippin
Keyword(s):  
Lebesgue Measure ◽  
Lower Bounds ◽  
Convex Sets ◽  
Upper And Lower Bounds ◽  
Dirichlet Eigenvalue ◽  
Torsion Function ◽  
Open Set ◽  
First Dirichlet Eigenvalue ◽  
Finite Lebesgue Measure

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)∕(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of ℝd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

A STRANGE ATTRACTOR WITH LARGE ENTROPY IN THE UNFOLDING OF A LOW RESONANT DEGENERATE HOMOCLINIC ORBIT

2006 ◽  
Vol 16 (12) ◽  
pp. 3509-3522 ◽  
Author(s):  
M. MARTENS ◽  
V. NAUDOT ◽  
J. YANG
Keyword(s):  
Vector Field ◽  
Lebesgue Measure ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Strange Attractor ◽  
Linear Part ◽  
Unperturbed System ◽  
Positive Lebesgue Measure ◽  
Open Set ◽  
Inclination Flip

The unfolding of a vector field exhibiting a degenerate homoclinic orbit of inclination-flip type is studied. The linear part of the unperturbed system possesses a resonance but the coefficient of the corresponding monomial vanishes. We show that for an open set in the parameter space, the system possesses a suspended cubic Hénon-like map. As a consequence, strange attractors with entropy close to log 3 persist in a positive Lebesgue measure set.

scholarly journals On the irregular points for systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2108-2131 ◽  
Author(s):  
YIWEI DONG ◽  
PIOTR OPROCHA ◽  
XUETING TIAN
Keyword(s):  
Lebesgue Measure ◽  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Zero Entropy ◽  
Time Average

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that satisfies the shadowing property, then the set of irregular points (i.e., points with divergent Birkhoff averages) has full entropy. Using this fact, we prove that, in the class of $C^{0}$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

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