Chaotic sets of continuous and discontinuous maps

This paper discusses Li-Yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps. Scrambled sets and maximal scrambled sets are introduced to characterize Li-Yorke chaotic sets. The orbit invariant for a scrambled set is discussed. Some properties about maximality, equivalence and uniqueness of maximal scrambled sets are also discussed. It is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps, such as the Gauss map, interval exchange transformations, and a class of planar piecewise isometries, are studied. Finally, some open problems on scrambled sets are listed and remarked.     

Thick attractors of step skew products

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.     

具有稠密临界轨道和预不动临界轨道的单峰映射的丰富性

证明了对于实二次族在参数空间存在正Ｌｅｂｅｓｇｕｅ测度集合Ｅ,使得Ｅ中几乎所有的参数,相应的映射在不变测度的支集上具有稠密的临界轨道;还证明了Ｅ中存在稠密集合使得相应映射的临界轨道进入它的反向不动点。     

轨道逼近时间集的密度

任意给定0pq1,证明了在符号系统中(进而在帐篷映射中)存在Mycielski集C,使得C中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.构造了线段上的连续映射,使其具有一个满Lebesgue测度的Mycielski集S,使得S中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.     

Syndetically proximal pairs

For continuous self-maps of compact metric spaces, we study the syndetically proximal relation, and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small. We show that any interval map f with positive topological entropy has a syndetically scrambled Cantor set, and an uncountable syndetically scrambled set invariant under some power of f. In the process of proving this, we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m?2, then there is n∈N such that the one-sided full shift on m symbols is topologically conjugate to a subsystem of fn² (the classical result gives only semi-conjugacy).     

CHAOS AND TOPOLOGICAL MIXING OF TREE MAPS

In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.     

Distributional chaos in a sequence

In this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence. As an application, it is proved that a continuous map of an interval is chaotic in the Li–Yorke sense if and only if it is distributively chaotic in a sequence.     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

An Exceptional Set in the Ergodic Theory of Markov Maps of the Interval

It is known that a Markov map T of the unit interval preservesa measure µ, say, equivalent to Lebesgue measure, andthat almost every point of the interval has a forward orbitunder T that is uniformly distributed with respect to µ.In the opposite direction the main result of this paper statesthat there is a set of points having Hausdorff dimension 1 whoseforward orbits are in a certain sense very far from being sodistributed. 1991 Mathematics Subject Classification: 58F08,28A80.     

Interval exchange transformations and some special flows are not mixing

An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds. The author is partially supported by grant NSF MCS 78-15278.     

Differentiability properties of weak solutions of certain variational problems in the theory of perfect elastoplastic plates

In this paper we deal with the regularity of weak solutions of some variational problems arising in the theory of perfect elastoplastic plates. All results concern differential properties of the tensor of moments, which is the solution of the dual variational problem. We show that the tensor of moments has generalized derivatives of first order which are locally square summable and prove that an open set exists where the solution is regular (the tensor of moments is Holder continuous) and some quadratic form of moments is less than a critical value. In the complement of this set the quadratic form is equal to the critical value almost everywhere. Under some additional assumptions the Lebesgue measure of the complement is zero and we have the regularity of a weak solution on the open set of full measure.     

Non-uniformly expanding dynamics in maps with singularities and criticalities

We investigate a one-parameter family of interval maps arising in the study of the geometric Lorenz flow for non-classical parameter values. Our conclusion is that for all parameters in a set of positive Lebesgue measure the map has a positive Lyapunov exponent. Furthermore, this set of parameters has a density point which plays an important dynamic role. The presence of both singular and critical points introduces interesting dynamics, which have not yet been fully understood.     

On the simplicity of the full group of ergodic transformations

LetE denote an invertible, non-singular, ergodic transformation of (0, 1). Then the full group ofE is perfect. IfE preserves the Lebesgue measure, then the full group is simple. IfE preserves no measure equivalent to Lebesgue, then the full group is simple. IfE preserves an infinite measure, then there exists a unique normal subgroup. IfT is any invertible transformation preserving the Lebesgue measure, then the full group is simple if and only ifT is ergodic on its support.     

β-变换下攀援集和distal集的测度性质

设β1为实数,T_β为[0,1]的β变换.攀援集的任何两个点随着时间的转移会越来越接近但同时又总能在任意长时间后保持一定的距离.证明了在Lebesgue测度意义下关于T_β的攀援集是一个零测集.Distal点对的两个点表示随着时间的转移始终保持着一定的距离.如果固定其中一个点x_0,所有满足x∈[0,1)且lim inf n→∞|T_β~n(x)-T_β~n(x_0)|0的点称为关于x_0的distal集,如果把这个集合记为R_β(x_0),根据Borel-Cantelli引理得到R_β(x_0)的Lebesgue测度为零.     

How good is lebesgue measure?

So, what is the answer to the question “How good is Lebesgue measure?” In the class of invariant measures, Lebesgue measure seems to be the best candidate to be a canonical measure. In the class of countably additive not necessarily invariant measures, to find a universal measure we have to use a strong additional set-theoretical assumption and this seems to be too high a price. Thus the best improvement of Lebesgue measure seems to be the Banach construction of a finitely additive isometrically invariant extension of Lebesgue measure on the plane and line. However, such a measure does not exist on Rⁿ for n ≤ 3, and to keep the theory of measures uniform for all dimensions we cannot accept the Banach measure on the plane as the best solution to the measure problem. From this discussion it seems clear that there is no reason to depose Lebesgue measure from the place it has in modern mathematics. Lebesgue measure also has a nice topological property called regularity: for every EL and every ɛ > 0, there exists an open set V⊃E and closed set F ⊂ E such that m(V/F) < ɛ. It is not difficult to prove that Lebesgue measure is the richest countably additive measure having this property (see [Ru], Thm. 2.20, p. 50).     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

On Variational Measures Related to Some Bases

We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann–Feller basis.     

Distributional chaos and spectral decomposition on the circle

Schweizer and Smı́tal [Trans. Amer. Math. Soc. 344 (1994) 737–754] introduced the notion of distributional chaos for continuous maps of the interval. In this paper we show that similar results, up to natural modifications, are valid for the continuous mappings of the circle. Thus any such map has a finite spectrum, which is generated by the map restricted to a finite collection of basic sets, and any scrambled set in the sense of Li and Yorke has a decomposition into three subsets (on the interval into two subsets) such that the distribution function generated on any such subset is bounded from below by a distribution function from the spectrum. While the results are similar, the original argument is not applicable directly and needs essential modifications.     

Specification on the interval

We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the -transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133-134) (for which we give a proof).

     

Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle

Avila recently introduced a new method for the study of the discrete Schrödinger operator with limit-periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit-periodic Verblunsky coefficients. Specifically, I represent these two-sided Verblunsky coefficients as a continuous sampling of the orbits of a Cantor group by a minimal translation. I then investigate the measures that arise on the unit circle as I vary the sampling function. I show that generically the spectrum is a Cantor set and we have empty point spectrum. Furthermore, there exists a dense set of sampling functions for which the corresponding spectrum is a Cantor set of positive Lebesgue measure, and all corresponding spectral measures are purely absolutely continuous.