Spectral Properties of Interval Exchange Maps

 In [7], Nogueira and Rudolph proved that for irreducible permutations not of rotation class almost every (a.e.) interval exchange transformation (i.e.t.) is topological weak mixing. It is conjectured that the claim holds if topological weak mixing is replaced by weak mixing. Here we study the behaviour of eigenfunctions of i.e.t. Our analysis gives alternative proofs of results due to Katok and Stepin [4] and Veech [10]: for certain permutations a.e. i.e.t. is weak mixing and for irreducible permutations a.e. i.e.t. is totally ergodic. (Received 1 February 2001)     

Linear Maps Preserving Invertibility or Related Spectral Properties

We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrumpreserving elementary operators. Several open problems are also mentioned.     

联系区间映射与圆周映射的函数方程

在解释区间映射和圆周映射的数量普适性现象的理论中，有关的重整化群函数方程及其解起着关键作用。本文研究具有广泛意义的联系这两种映射的函数方程得到的主要结果是它存在无穷多个Ｃ￣∞解，且可用构造性方法给出，其参数的范围比［９］中的Ｄ大。最后还提出一些值得进一步研究的问题。     

关于区间上的Lorenz映射

本文引进了变换ζ．首先，用比文［４］简捷的方法证明了Ｌｏｒｅｎｚ映射的Ｓａｒｋｏｖｓｋｉ定理并更正了文［４］中的一个错误，还得到了扩张的Ｌｏｒｅｎｚ映射中周期点的存在性与其扩张常数之间的关系．其次，证明了Ｌｏｒｅｎｚ映射与其在变换ζ的像具有相等的捏制行列式．此外，给出了广义的Ｌｏｒｅｎｚ映射的周期点的存在性适合Ｓａｒｋｏｖｓｋｉ序关系的一个充分条件     

具有进位吸引子的区间映射

对于一个具有n-进位吸引子的区间连续映射,证明了：“n不是2的方幂”是该映射具有正拓扑熵的充分条件但不是必要条件；探讨了函数方程f~3(λx)=λf(x)的一类解,并证明这类解中的每一个成员都有3-进位吸引子．     

具有进位吸引子的区间映射

对于一个具有n-进位吸引子的区间连续映射,证明了"n不是2的方幂"是该映射具有正拓扑熵的充分条件但不是必要条件;探讨了函数方程f3(λx)=λf(x)的一类解,并证明这类解中的每一个成员都有3-进位吸引子.     

On C*-Algebras from Interval Maps

Given a unimodal interval map f, we construct partial isometries acting on Hilbert spaces associated to the orbit of each point. Then we prove that such partial isometries give rise to representations of a C*-algebra associated to the subshift encoding the kneading sequence of the critical point. This construction has the advantage of incorporating maps with a non necessarily Markov partition (e.g. Fibonacci unimodal map). If we are indeed in the presence of a finite Markov partition, then we prove that these new representations coincide with the (previously considered by the authors) representations arising from the Cuntz–Krieger algebra of the underlying (finite) transition matrix.     

On Iterated Maps of the Interval with Holes

A kneading theory is generalized to maps of the interval with several discontinuity points and holes. Alternative methods to evaluate topological entropy are introduced and related. Also we study the parametrization of families of maps with holes and the monotonicity properties of the topological entropy.     

保持谱半径的乘法映射

设X和Y为无限维Banach空间,Φ：B（X）→B（Y）是保持谱半径的满射,且秩为1算子,则Φ具有形式Φ（T）=ATA∧-1,这里A：X→Y或是线性拓扑同构映射或是线性拓扑同构映射的共轭。     

保持算子乘积谱函数的映射

设A和B为无限维复Banach空间上的标准算子代数,记ΔR(·)为下列谱函数之一σR(·),σRl(·),σRr(·),σRl(·)∩σRr(·),(a)σR(·),ησR(·),σRp(·),σRc(·),σRap(·),σRs(·),σRap(·)∩σRs(·),σRp(·)∩σRc(·),σRp(·)∪σRc(·),其中R=A或B.证明了A和B之间的每个保持算子Jordan三乘积(算子乘积)之谱函数ΔR(·)的满射φ必有形式φ=επ,其中ε是1的立方根(1的平方根)而π或者是A和B之间的代数同构,或者是代数反同构.也获得不定度规空间上的标准算子代数之间保持算子斜乘积之谱函数的映射的完全刻画.     

保持算子乘积谱函数的映射

设A和B为无限维复Banach空间上的标准算子代数,记Δ~R(·)为下列谱函数之一:σ~R(·),σ_l~R(·),σ_r~R(·),σ_l~R(·)∩σ_r~R(·),(?)σ~R(·),(?)σ~R(·),σ_p~R(·),σ_c~R(·),σ_(ap)~R(·),σ_s~R(·),σ_(ap)~R(·)∩σ_s~R(·),σ_p~R(·)∩σ_c~R(·),σ_p~R(·)∪σ_c~R(·),其中R=A或B.证明了A和B之间的每个保持算子Jordan三乘积(算子乘积)之谱函数△~R(·)的满射Φ必有形式Φ=(?)π,其中(?)是1的立方根(1的平方根)而π或者是A和B之间的代数同构,或者是代数反同构.也获得不定度规空间上的标准算子代数之间保持算子斜乘积之谱函数的映射的完全刻画.     

Anosov映射的性质

设f是紧致度量空间M上的自覆盖映射,本文证明(a)?(c)结合[2](a)?(b)且(a)?(c)     

单峰扩张映射的捏制序列

设Ｐ及ＡＣ均是准素序列并满足ｍｉｎ｛Ｐ，ＡＣ｝ＲＬＲ∞，ρ（Ｐ）及ρ（ＡＣ）分别是Ｐ及ＡＣ的特征值．设ｆ∈Ｃ０（Ｉ，Ｉ）是个单峰扩张映射并具有扩张常数λ，ｍ是个非负整数．本文证明了若λ（ρ（Ｐ））１／２ｍ，则ｆ的捏制序列Ｋ（ｆ）（ＲＣ）ｍＰ；若λ＞（ρ（ＡＣ））１／２ｍ，则对任极大序列Ｅ，Ｋ（ｆ）＞（ＲＣ）ｍＡＣＥ．（ρ（Ｐ））１／２ｍ及（ρ（ＡＣ））１／２ｍ均是下述意义下的最佳值，即若其中任一个被较小的值代替，则相应的结论便不成立．     

线段映射的局部变差增长与局部拓扑熵

本文考虑闭区间上变差有界的连续映射f:I→I的局部变差增长γ(x,f)与局部拓扑熵h(x,f)．将证明γ(x,f)≥h(x,f)对所有x∈I成立,并且局部变差增长映射γf(x)=γ(x,f)与局部拓扑熵映射sf(x)=h(x,f)都是上半连续的,得到一个变分原理:局部变差增长γ(x,f)与局部拓扑熵h(x,f)的上确界分别等于全局变差增长γ(f)=limn→∞1/nln Var(fn)与拓扑熵h(f)．当映射f:I→I拓扑传递时,与Brin 和Katok对局部(测度)熵的讨论类似,我们证明,至多除一个不动点外,局部变差增长γ(x,f)与局部拓扑熵h(x,f)在开区间I°内恒为常值．     

Statistical Properties of Unimodal Maps

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.