scholarly journals Bernoulli diffeomorphisms with n − 1 non-zero exponents

1981 ◽  
Vol 1 (1) ◽  
pp. 1-7 ◽  
Author(s):  
M. Brin
Keyword(s):  
Lebesgue Measure ◽  
Skew Product ◽  
Anosov Flow ◽  
Characteristic Exponents ◽  
Zero Characteristic ◽  
Almost Everywhere

AbstractFor every manifold of dimension n ≥ 5 a diffeomorphism f which has n − 1 non-zero characteristic exponents almost everywhere is constructed. The diffeomorphism preserves the Lebesgue measure and is Bernoulli with respect to this measure. To produce this example a diffeomorphism of the 2-disk is extended by means of an Anosov flow, and this skew product is embedded in ℝn.

scholarly journals Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

1982 ◽  
Vol 2 (3-4) ◽  
pp. 439-463 ◽  
Author(s):  
Feliks Przytycki
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Bifurcation Parameter ◽  
Two Dimensional ◽  
Open Area ◽  
Dimensional Torus ◽  
Characteristic Exponents ◽  
Almost Everywhere ◽  
Lyapunov Characteristic Exponents ◽  
Real Analytic

AbstractWe find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

scholarly journals A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.

scholarly journals Uniform almost everywhere domination

2006 ◽  
Vol 71 (3) ◽  
pp. 1057-1072 ◽  
Author(s):  
Peter Cholak ◽  
Noam Greenberg ◽  
Joseph S. Miller
Keyword(s):  
Lebesgue Measure ◽  
Almost Everywhere ◽  
A Priority ◽  
Almost All

AbstractWe explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

On Continuous Functions which are Partially Differentiable Almost Everywhere

1969 ◽  
Vol 12 (5) ◽  
pp. 668-672
Author(s):  
L.V. Toralballa
Keyword(s):  
Lebesgue Measure ◽  
Surface Area ◽  
Measure Zero ◽  
Continuous Functions ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.

On the almost sure coverage property of Voronoi tessellation: the ℝ1 case

2001 ◽  
Vol 33 (4) ◽  
pp. 756-764 ◽  
Author(s):  
Estate Khmaladze ◽  
N. Toronjadze
Keyword(s):  
Lebesgue Measure ◽  
Law Of Large Numbers ◽  
Voronoi Tessellation ◽  
Measurable Subset ◽  
Positive Answer ◽  
Jump Point ◽  
Strong Law ◽  
Almost Everywhere ◽  
Large Numbers ◽  
Nonhomogeneous Poisson Processes

This paper raises the following question: let {Φn(A), A ⊂ ℝd} be a Poisson process with intensity nf(x), x ∈ ℝd and let c(Xi | Φn) be a Voronoi tile with nucleus Xi (a jump point of Φn). Let μ(.) denote Lebesgue measure in ℝd. Is it true that, for any bounded measurable subset B of ℝd, ∑Xi∈Bμ(c(Xi| Φn)) → μ(B) almost surely as n → ∞ only if f > 0 almost everywhere? This statement can be viewed as the strong law of large numbers for Voronoi tessellation. Though the positive answer may seem ‘obvious’, we could not find any such statement, especially for arbitrary measurable B and nonhomogeneous Poisson processes. For B with the boundary of Lebesgue measure 0 the proof is simple. We prove in this paper that the statement is true for ℝ1.

On the almost sure coverage property of Voronoi tessellation: the ℝ1 case

2001 ◽  
Vol 33 (04) ◽  
pp. 756-764
Author(s):  
Estate Khmaladze ◽  
N. Toronjadze
Keyword(s):  
Lebesgue Measure ◽  
Law Of Large Numbers ◽  
Voronoi Tessellation ◽  
Measurable Subset ◽  
Positive Answer ◽  
Jump Point ◽  
Strong Law ◽  
Almost Everywhere ◽  
Large Numbers ◽  
Nonhomogeneous Poisson Processes

This paper raises the following question: let {Φ n (A), A ⊂ ℝ d } be a Poisson process with intensity nf(x), x ∈ ℝ d and let c(X i | Φ n ) be a Voronoi tile with nucleus X i (a jump point of Φ n ). Let μ(.) denote Lebesgue measure in ℝ d . Is it true that, for any bounded measurable subset B of ℝ d , ∑ X i ∈B μ(c(X i | Φ n )) → μ(B) almost surely as n → ∞ only if f > 0 almost everywhere? This statement can be viewed as the strong law of large numbers for Voronoi tessellation. Though the positive answer may seem ‘obvious’, we could not find any such statement, especially for arbitrary measurable B and nonhomogeneous Poisson processes. For B with the boundary of Lebesgue measure 0 the proof is simple. We prove in this paper that the statement is true for ℝ1.

scholarly journals Classification of partially hyperbolic diffeomorphisms under some rigid conditions

2020 ◽  
pp. 1-12
Author(s):  
PABLO D. CARRASCO ◽  
ENRIQUE PUJALS ◽  
FEDERICO RODRIGUEZ-HERTZ
Keyword(s):  
Three Dimensional ◽  
Skew Product ◽  
Anosov Diffeomorphism ◽  
Anosov Flow ◽  
Hyperbolic Diffeomorphisms ◽  
Finite Covers ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

Conjugation between circle maps with several break points

2015 ◽  
Vol 36 (8) ◽  
pp. 2351-2383 ◽  
Author(s):  
ABDELHAMID ADOUANI
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Function ◽  
Absolutely Continuous ◽  
Circle Maps ◽  
Almost Everywhere ◽  
Rotation Numbers ◽  
Break Points

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.

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