A characterization of hyperbolic rational maps

We give a topological characterization of rational maps with disconnected Julia sets. Our results extend Thurston’s characterization of postcritically finite rational maps. In place of iteration on Teichmüller space, we use quasiconformal surgery and Thurston’s original result.     

A characterization of adding machine maps

Let f :X→X be a continuous map of a compact metric space to itself. We prove that f is topologically conjugate to an adding machine map if and only if X is an infinite minimal set for f and each point of X is regularly recurrent. Moreover, if X is an infinite minimal set for f and one point of X is regularly recurrent, then f is semiconjugate to an adding machine map.     

A characterization of hyperbolic potentials of rational maps

Consider a rational map f of degree at least 2 acting on its Julia set J(f), a H?lder continuous potential φ: J(f) → ℝ and the pressure P(f,φ). In the case where

A characterization of non-linear maps satisfying orthogonality properties

Maps not necessarily linear but monotone (order preserving) between function spaces are analyzed. Characterizations of maps T from functions on X to those on Y with the property that the image Tf(y) depends on the value of f at one point \(x\in X\) are established. Then T has a functional representation, namely, Tf(y) is equal to a function \(F_y\) composed with f(x). In particular, for X extremally disconnected, T satisfies the above property for non-negative functions on X if and only if is finitely disjointness preserving (\(\wedge f_i=0 \Rightarrow \wedge T(f_i)=0\)), orthogonally additive (\(f\wedge g=0 \Rightarrow T(f+g) = T(f) + T(g)\)), and satisfies a continuity condition. In the absence of continuity conditions, the above order theoretic conditions are equivalent to a local condition, specifically, \(Tf(y)=Tg(y)\) whenever \(f = g\) on a neighborhood of x. Results for more general domains are provided as well as consequences for bijections.     

Linear maps preserving the essential spectral radius

Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert H. We characterize linear maps from L(H) onto itself that preserve the essential spectral radius.     

A characterization of inverse limits of Q-bundle maps

In this paper we prove that a surjection between metric compacta is a hereditary shape equivalence if and only if it is an inverse limit of trivial Q-bundle maps. This result was conjectured by T.B. Rushing. Near-homeomorphisms are instrumental to the proof.     

Commuting holomorphic maps on the spectral unit ball

We prove that if F is a holomorphic map from the open spectralunit ball of a primitive Banach algebra into itself satisfyingF(0) = 0, F^' (0) = I and F(x) x = xF(x) for every x, then Fis the identity map. Using this, we prove that if is a semisimpleBanach algebra and is a primitive Banach algebra, then anyunital spectral isometry from onto which locally preservescommutativity is a Jordan morphism. The same is true when and are both assumed to be von Neumann algebras.     

A characterization of positively decomposable non-linear maps between Banach lattices

A map between Banach lattices E and F is called positively decomposable if Tf = g ₁ + g ₂ for f, g ₁, g ₂ positive and g ₁ and g ₂ disjoint implies there exist disjoint positive elements f ₁ and f ₂ each less than f with the property that Tf ₁ = g ₁ and Tf ₂ = g ₂. Recently, the positive decomposability of linear Carleman operators on Banach lattices were characterized using disjointness condition of images of the approximate atoms. This note provides an extension of the characterization for a class of non-linear maps. Further, disjointness preserving maps are studied.      

A note on the spectral characterization of dumbbell graphs

The dumbbell graph, denoted by D_a,b,c, is a bicyclic graph consisting of two vertex-disjoint cycles C_a and C_b joined by a path P_c+3 (c-1) having only its end-vertices in common with the two cycles. By using a new cospectral invariant for (r,r+1)-almost regular graphs, we will show that almost all dumbbell graphs (without cycle C₄ as a subgraph) are determined by the adjacency spectrum.     

Combinatorial characterization of sub-hyperbolic rational maps

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps among post-critically finite branched coverings of the two sphere to itself. A completed proof was written by Douady and Hubbard in their paper [A. Douady, J.H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993) 263-297]. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps among sub-hyperbolic semi-rational branched coverings of the two sphere to itself. The goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub-hyperbolic rational maps by adapting some arguments in the proof in Douady and Hubbard's paper.     

Additive maps preserving operators of inner local spectral radius zero

Let \({\mathcal {B}}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space \(X\) . We determine the form of surjective additive maps \(\varphi :{\mathcal {B}}(X)\rightarrow {{\mathcal {B}}(X)}\) which preserve operators of inner local spectral radius zero at points of \(X\) .     

Another characterization of convexity for set-valued maps

We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative of F which is constructed with the help of an abstract subdifferential notion of Penot . A detailed discussion is devoted to special cases when the contingent, the Fréchet and the Clarke-Rockafellar subdifFerentials Sixe used as this abstract subdifferential.     

On functors preserving skeletal maps and skeletally generated compacta

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{?1}(A)$ of each nowhere dense subset $A?Y$ is nowhere dense in X. We prove that a normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map $f:X\to Y$ between metrizable zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{?1}(f(x))|>1\}$ the map $Ff:FX\to FY$ is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ preserves the class of skeletally generated compacta. This contrasts with the known ??epin?s result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.     

A spectral characterization of the uniform continuity of strongly continuous groups

Let X be a Banach space and a strongly continuous group of linear operators on X. Set and where is the unit circle and denotes the spectrum of T(t). The main result of this paper is: is uniformly continuous if and only if is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007