Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps

We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy.?For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case. Oblatum 2-IV-2002 & 2-V-2002?Published online: 6 August 2002 RID="*" ID="*"All authors are supported by the European Science Foundation program PRODYN. The first author is also supported by the Foundation for Polish Sciences and Polish KBN grant 2P03A 00917. The second author is grateful to IMPAN and KTH for hospitality and is also supported by a Polish-French governmental agreement, Fundacion Andes and a “Beca Presidente de la Republica,” Chile. The third author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.     

Non-uniform hyperbolicity and universal bounds for S-unimodal maps

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors. Oblatum 4-VII-1996 & 4-VII-1997     

Collet, Eckmann and Hölder

We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical orbits, yields the H?lder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected, and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least one non-empty fully invariant Fatou component. Oblatum 22-III-1996 & 15-VII-1997     

Möbius disjointness for topological models of ergodic measure-preserving systems with quasi-discrete spectrum

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that Sarnak's Möbius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.     

On the metrical properties of the configuration space

We study some topological and metrical properties of configuration spaces. In particular, we introduce a family of metrics on the configuration space Γ, which makes it a Polish space. Compact functions on Γ are also considered. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Obtaining fast error rates in nonconvex situations

We show that under mild assumptions on the learning problem, one can obtain a fast error rate for every reasonable fixed target function even if the base class is not convex. To that end, we show that in such cases the excess loss class satisfies a Bernstein type condition.     

The large deviations theorem and ergodicity

In this paper, some relationships between stochastic and topological properties of dynamical systems are studied. For a continuous map f from a compact metric space X into itself, we show that if f satisfies the large deviations theorem then it is topologically ergodic. Moreover, we introduce the topologically strong ergodicity, and prove that if f is a topologically strongly ergodic map satisfying the large deviations theorem then it is sensitively dependent on initial conditions.     

Harmonic measure and expansion on the boundary of the connectedness locus

The paper develops a technique for proving properties that are typical in the boundary of the connectedness locus with respect to the harmonic measure. A typical expansion condition along the critical orbit is proved. This condition implies a number of properties, including the Collet-Eckmann condition, Hausdorff dimension less than 2 for the Julia set, and the radial continuity in the parameter space of the Hausdorff dimensions of totally disconnected Julia sets. Oblatum 6-XI-1998 & 12-V-2000?Published online: 11 October 2000     

Improvement of Hartman''''s linearization theorem

Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) is bounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphism of Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x = Ax. In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we prove the result of global topological linearization without any special limitation and adding any condition. Thus, Hartman's linearization theorem is improved essentially.     

G2,n中的可兼共形调和曲面

本文通过建立复Ｇｒａｓｓｍａｎｎ流形Ｇ２，ｎ中调和曲面的Ｇａｕｓｓ丛的基本公式，在适当的拓扑条件下给出了Ｇ２，ｎ中可兼共形调和曲面的构造定理．推广和改进了Ｂｕｒｓｔａｌｌ和Ｗｏｏｄ的低亏格曲面的结果．     

Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography and Codes

This contribution is concerned with a generalization of Itoh and Tsujii's algorithm for inversion in extension fields . Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard basis representation and relevant complexity expressions are established, consisting of the number of extension field multiplications and exponentiations. As the main contribution, for three important classes of fields we show that the Frobenius map can be explored to perform the exponentiations required for the inversion algorithm efficiently. As an important consequence, Itoh and Tsujii's inversion method shows almost the same practical complexity for standard basis as for normal basis representation for the field classes considered.     

Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.     

Unique Ergodicity for Zero-entropy Dynamical Systems with the Approximate Product Property

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.     

Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup

We study the controllability problem for a system governed by a semilinear differential inclusion in a Banach space not assuming that the semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows us to apply the topological degree theory for condensing operators and to obtain the controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity. As application we consider the controllability for a system governed by a perturbed wave equation.     

Geometric representation of substitutions of Pisot type

We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.

     

伪单调映射的拓扑度及应用

利用S+型映射的拓扑度,导出了伪单调映射紧扰动的拓扑度,并讨论了该拓扑度对算子值域的应用．     

The Cauchy problem about Dirac-wave map from the 2-dimension Minkowski space to a complete Riemannian manifold

When a target manifold is complete with a bounded curvature, we prove that there exists a unique global solution which satisfies the Euler-lagrange equation of for the given Cauchy data.     

Distributive Lattices with a Negation Operator

In this note we introduce and study algebras (L, V, Λ, ?, 0,1) of type (2, 2,1,1,1) such that (L, V, ?, 0,1) is a bounded distributive lattice and ? is an operator that satisfies the condition ? (a V b) = a ? b and ? 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.