Displaying similar documents to “Parametrized Cichoń's diagram and small sets”

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

Similarity:

We study ergodicity of cylinder flows of the form    T f : T × T × , T f ( x , y ) = ( x + α , y + f ( x ) ) , where f : T is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that D k f is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of D k f have some good properties, then T f is ergodic. Moreover, there exists ε f > 0 such that if v : T is a function with zero integral such that D k v is of bounded...

A note on Tsirelson type ideals

Boban Veličković (1999)

Fundamenta Mathematicae

Similarity:

Using Tsirelson’s well-known example of a Banach space which does not contain a copy of c 0 or l p , for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P ( ) / I T and P ( ) / I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ ℕ such that n A 1 / ( n + 1 ) < . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

Similarity:

We prove the following theorem: Given a⊆ω and 1 α < ω 1 C K , if for some η < 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Co-H-structures on equivariant Moore spaces

Martin Arkowitz, Marek Golasiński (1994)

Fundamenta Mathematicae

Similarity:

Let G be a finite group, 𝕆 G the category of canonical orbits of G and A : 𝕆 G 𝔸 b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with E x t n - 1 ( A , A A ) . Then the case G = p k leads to an example of...

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

Similarity:

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in A ( ( X ) ) into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential...

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

Similarity:

For X ⊆ [0,1], let D X denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of D X . For α ≥ 3, X is properly D ξ ( Π α 0 ) iff D X is properly D ξ ( Π 1 + α 0 ) . We also show that for every nonempty set X ⊆[0,1], D X is Π 3 0 -hard. For each nonempty Π 2 0 set X ⊆ [0,1], in particular for X = x, D X is Π 3 0 -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is Π 3 0 -complete. Moreover,...

Growth of the product j = 1 n ( 1 - x a j )

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

Similarity:

We estimate the maximum of j = 1 n | 1 - x a j | on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when a j is j k or when a j is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when a j is j.    In contrast we show, under fairly general conditions, that the maximum is less than 2 n / n r , where r is an arbitrary positive number. One consequence...

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

Similarity:

Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and α < ω 1 then f ○ g ∈ B α ( Z ) for every g B α ( Z ) Z I if and only if f is continuous on I, where B α ( Z ) stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes S α ( Z ) ( α > 0 ) in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally...

A note on strange nonchaotic attractors

Gerhard Keller (1996)

Fundamenta Mathematicae

Similarity:

For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ T 1 × + with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties:  1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in T 1 × + . The set θ:ϕ(θ) ≠ 0 is meager but has full 1-dimensional Lebesgue measure.  2. The omega-limit of Lebesgue-a.e point in T 1 × + is Γ ̅ , but for a residual set of points in T 1 × + the omega...

On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

Similarity:

The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping f : S k k there exists a point x S k such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which S k is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of i n f x | | f ( x ) - f ( - x ) | | which were previously known (as far as the author knows) only for f linear (cf. [1]).

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

Similarity:

Let X 2 w . Consider the class of all Borel F X × 2 w with null vertical sections F x , x ∈ X. We show that if for all such F and all null Z ⊆ X, x Z F x is null, then for all such F, x X F x 2 w . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Ergodic averages and free 2 actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

Similarity:

If the ergodic transformations S, T generate a free 2 action on a finite non-atomic measure space (X,S,µ) then for any c 1 , c 2 there exists a measurable function f on X for which ( N + 1 ) - 1 j = 0 N f ( S j x ) c 1 and ( N + 1 ) - 1 j = 0 N f ( T j x ) c 2 µ -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Hisao Kato (1994)

Fundamenta Mathematicae

Similarity:

A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c (resp. d i a m f n ( A ) > c ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate...

The dimension of X^n where X is a separable metric space

John Kulesza (1996)

Fundamenta Mathematicae

Similarity:

For a separable metric space X, we consider possibilities for the sequence S ( X ) = d n : n where d n = d i m X n . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is X n such that S ( X n ) = n , n + 1 , n + 2 , . . . , Y n , for n >1, such that S ( Y n ) = n , n + 1 , n + 2 , n + 2 , n + 2 , . . . , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of 2 is shown to exist which satisfies 1 = d i m X = d i m X 2 and d i m X 3 = 2 .

On the restricted Waring problem over 2 n [ t ]

Luis Gallardo (2000)

Acta Arithmetica

Similarity:

1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, 16 , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is...

Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik (1998)

Fundamenta Mathematicae

Similarity:

Let α be an ergodic rotation of the d-torus 𝕋 d = d / d . For any piecewise smooth function f : 𝕋 d with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on L 2 ( 𝕋 d ) is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product S f : 𝕋 d + 1 𝕋 d + 1 must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum...

A complement to the theory of equivariant finiteness obstructions

Paweł Andrzejewski (1996)

Fundamenta Mathematicae

Similarity:

It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family w α H ( X ) of elements of the groups K 0 ( [ π 0 ( W H ( X ) ) α * ] ) . We prove that every family w α H of elements of the groups K 0 ( [ π 0 ( W H ( X ) ) α * ] ) can be realized as the family of equivariant finiteness obstructions w α H ( X ) of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s...

Partition properties of subsets of Pκλ

Masahiro Shioya (1999)

Fundamenta Mathematicae

Similarity:

Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any f : n < ω [ X ] n γ with X P κ λ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with | f ' ' [ Y ] n | = 1 for any n < ω.