Displaying similar documents to “Strong bifurcation loci of full Hausdorff dimension”

The supports of higher bifurcation currents

Romain Dujardin (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let ( f λ ) λ Λ be a holomorphic family of rational mappings of degree d on 1 ( ) , with k marked critical points c 1 , ... , c k . To this data is associated a closed positive current T 1 T k of bidegree ( k , k ) on Λ , aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c 1 , ... , c k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp ( T 1 T k ) . ...

On bifurcation and uniqueness results for some semilinear elliptic equations involving a singular potential

Manuela Chaves, Jesús García-Azorero (2006)

Journal of the European Mathematical Society

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We present some results concerning the problem Δ u = λ u | x | 2 + u q , u > 0 in Ω , u | Ω = 0 , where 0 < q < ( N + 2 ) / ( N 2 ) , q 1 , λ 0 and Ω is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.

Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci, Roberto Livrea (2003)

Annales Polonici Mathematici

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We deal with a bifurcation result for the Dirichlet problem ⎧ - Δ p u = μ / | x | p | u | p - 2 u + λ f ( x , u ) a.e. in Ω, ⎨ ⎩ u | Ω = 0 . Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number λ * μ such that for every λ ] 0 , λ * μ [ the above problem admits a nonzero weak solution u λ in W 1 , p ( Ω ) satisfying l i m λ 0 | | u λ | | = 0 .

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ). ...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map g σ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set J 0 , σ is continuous at σ₀ as the function of the parameter σ ¯ if and only if H D ( J 0 , σ ) 4 / 3 . Since H D ( J 0 , σ ) > 4 / 3 on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of H D ( J 0 , σ ) on an open and dense subset of...

S -shaped component of nodal solutions for problem involving one-dimension mean curvature operator

Ruyun Ma, Zhiqian He, Xiaoxiao Su (2023)

Czechoslovak Mathematical Journal

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Let E = { u C 1 [ 0 , 1 ] : u ( 0 ) = u ( 1 ) = 0 } . Let S k ν with ν = { + , - } denote the set of functions u E which have exactly k - 1 interior nodal zeros in (0, 1) and ν u be positive near 0 . We show the existence of S -shaped connected component of S k ν -solutions of the problem u ' 1 - u ' 2 ' + λ a ( x ) f ( u ) = 0 , x ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter, a C ( [ 0 , 1 ] , ( 0 , ) ) . We determine the intervals of parameter λ in which the above problem has one, two or three S k ν -solutions. The proofs of the main results are based upon the bifurcation technique.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets J 0 , σ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of J 0 , σ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS ϕ σ , α n , k is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato, Eiichi Matsuhashi (2006)

Colloquium Mathematicae

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space C ( X , I p + 2 k + 1 - i ) such that the diagonal product f × g : X Y × I p + 2 k + 1 - i is an (i+1)-to-1 map is a dense G δ -subset of C ( X , I p + 2 k + 1 - i ) . In this paper, we prove that if f: X → Y is as above and D j (j = 1,..., k) are superdendrites, then the set of maps h in C ( X , j = 1 k D j × I p + 1 - i ) such that f × h : X Y × ( j = 1 k D j × I p + 1 - i ) is (i+1)-to-1 is a dense G δ -subset of C ( X , j = 1 k D j × I p + 1 - i ) for each 0 ≤ i ≤ p.

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials V μ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials V μ are defined by V μ ( z ) = B l o g ( 1 / | ϕ z ( w ) | ) d μ ( w ) , where for a fixed z ∈ B, ϕ z denotes the holomorphic automorphism of B satisfying ϕ z ( 0 ) = z , ϕ z ( z ) = 0 and ( ϕ z ϕ z ) ( w ) = w for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

The bicrossed products of H 4 and H 8

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let H 4 and H 8 be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through H 8 and H 4 (equivalently, any bicrossed product between the Hopf algebras H 8 and H 4 ) must be isomorphic to one of the following four Hopf algebras: H 8 H 4 , H 32 , 1 , H 32 , 2 , H 32 , 3 . The set of all matched pairs ( H 8 , H 4 , , ) is explicitly described, and then the associated bicrossed product is given by generators and relations.

Coloring Cantor sets and resolvability of pseudocompact spaces

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let us denote by Φ ( λ , μ ) the statement that 𝔹 ( λ ) = D ( λ ) ω , i.e. the Baire space of weight λ , has a coloring with μ colors such that every homeomorphic copy of the Cantor set in 𝔹 ( λ ) picks up all the μ colors. We call a space X π -regular if it is Hausdorff and for every nonempty open set U in X there is a nonempty open set V such that V ¯ U . We recall that a space X is called feebly compact if every locally finite collection of open sets in X is finite. A Tychonov space is pseudocompact if and...

R z -supercontinuous functions

Davinder Singh, Brij Kishore Tyagi, Jeetendra Aggarwal, Jogendra K. Kohli (2015)

Mathematica Bohemica

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A new class of functions called “ R z -supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of R z -supercontinuous functions properly includes the class of R cl -supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of cl -supercontinuous ( clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a zero-dimensional space and C c ( X ) be the set of all continuous real valued functions on X with countable image. In this article we denote by C c K ( X ) (resp., C c ψ ( X ) ) the set of all functions in C c ( X ) with compact (resp., pseudocompact) support. First, we observe that C c K ( X ) = O c β 0 X X (resp., C c ψ ( X ) = M c β 0 X υ 0 X ), where β 0 X is the Banaschewski compactification of X and υ 0 X is the -compactification of X . This implies that for an -compact space X , the intersection of all free maximal ideals in C c ( X ) is equal to C c K ( X ) , i.e., M c β 0 X X = C c K ( X ) . By applying...