Hyperbolic structure preserving isomorphisms of Markov shifts. II

This paper is a continuation of [14] and deals with metric isomorphisms of Markov shifts which are finitary and hyperbolic structure preserving. We prove that theβ-function introduced by S. Tuncel in [15] is an invariant of such isomorphisms. Following [5] this result is extended to Gibbs measures arising from functions with summable variation. Finally we prove that, for anyC ² Axiom A diffeomorphism on a basic set Ω, and for any equilibrium state associated with a Hölder continuous function on Ω, the Markov shifts arising from different Markov partitions of Ω are isomorphic via a finitary, hyperbolic structure preserving isomorphism. This fact leads to a rich class of examples of such isomorphisms (other examples are provided by finitary isomorphisms of Markov shifts with finite expected code lengths — cf. [14]).     

Constructing finitary isomorphisms with finite expected coding times

This paper is motivated by the question of whether the invariants β, Δ,cΔ completely characterize isomorphism of Markov chains by finitary isomorphisms that have finite expected coding times (fect). We construct a finitary isomorphism with fect under an additional condition. Whether coincidence of β, Δ,cΔ implies the required condition remains open.     

Conditional pressure and coding

We use pressure to obtain invariants for bounded-to-one block homomorphisms between Markov shifts. These invariants enable us to show that if there is a bounded-to-one block homomorphism between Bernoulli shifts given by probability vectorsp andq thenq may be obtained fromp by a permutation. The invariants may be viewed as conditional pressures; a convergence theorem for eigenmeasures of Ruelle operators motivates the definition of conditional pressure and helps establish our invariants for regular isomorphism of Markov shifts. It follows that Bernoulli shifts given by probability vectorsp andq are regularly isomorphic iffq is a permutation ofp. We employ our invariants also in the context of a finite equivalence. Finally we indicate that ratio variational principles yield further invariants.     

Code length between Markov processes

LetX ₁ andX ₂ be two mixing Markov shifts over finite alphabet. If the entropy ofX ₁ is strictly larger than the entropy ofX ₂, then there exists a finitary homomorphism ϕ:X ₁→X ₂ such that the code length is anL ^p random variable for allp<4/3. In particular, the expected length of the code ϕ is finite. Research supported by KBN grant 2 P03A 039 15 1998–2001.     

Universal finitary codes with exponential tails

In 1977, Keane and Smorodinsky showed that there exists a finitaryhomomorphism from any finite-alphabet Bernoulli process to anyother finite-alphabet Bernoulli process of strictly lower entropy.In 1996, Serafin proved the existence of a finitary homomorphismwith finite expected coding length. In this paper, we constructsuch a homomorphism in which the coding length has exponentialtails. Our construction is source-universal, in the sense thatit does not use any information on the source distribution otherthan the alphabet size and a bound on the entropy gap betweenthe source and target distributions. We also indicate how ourmethods can be extended to prove a source-specific version ofthe result for Markov chains.     

Non-realizability and ending laminations: Proof of the density conjecture

We give a complete proof of the Bers?CSullivan?CThurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.     

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids.     

Some metric invariants for Markov shifts

Summary Metric invariants based on the logarithmic scale of infinity are obtained for irreducible, infinite-measure-preserving Markov shifts, and are applied to three classes of Markov shifts.     

论附属于某种连续映像的不等式及其在纤维空间中的应用

<正> §1. 设 X,Y 为拓扑空间,又设 f:X→Y 为连续映像.J.H.C.Whitehead 证明 X,Y 为 CW 丛而 f 能导出基本群及上同调群间的同模对应时,f 为同伦对等映像.映像 f 是否存在,不仅与 X,Y 的基本群及上同调群的构造有关,而与 X,Y 内在的几何结构有密切的关系.连续照像 f 导出 X,Y 之间上同调群的准同模对应 f,那末 f 能与某些准同模对应相交换,由此 J.H.C.Whitehead 指出正则准同模的观念.由[4]可知正则同模论供应我们许多同伦不变量,它们是直接可以计算的东西,并且对于 X,Y 间连续映像的分类问题,应该有密切的关系.     

Finitarily deterministic generators for zero entropy systems

Zero entropy processes are known to be deterministic—the past determines the present. We show that each is isomorphic, as a system, to a finitarily deterministic one, i.e., one in which to determine the present from the past it suffices to scan a finite (of random length) portion of the past. In fact we show more: the finitary scanning can be done even if the scanner is noisy and passes only a small fraction of the readings, provided the noise is independent of our system. The main application we present here is that any zero entropy system can be extended to a random Markov process (namely one in which the conditional distribution of the present given the past is a mixture of finite state Markov chains). This allows one to study zero entropy transformations using a procedure completely different from the usual cutting and stacking.     

Projective deformations of hyperbolic Coxeter 3-orbifolds

By using Klein??s model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev??s theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.     

Multirectangular invariants for power Köthe spaces

Using some new linear topological invariants, isomorphisms and quasidiagonal isomorphisms are investigated on the class of first type power Köthe spaces [Proceedings of 7th Winter School in Drogobych, 1976, pp. 101-126; Turkish J. Math. 20 (1996) 237-289; Linear Topol. Spaces Complex Anal. 2 (1995) 35-44]. This is the smallest class of Köthe spaces containing all Cartesian and projective tensor products of power series spaces and closed with respect to taking of basic subspaces (closed linear hulls of subsets of the canonical basis). As an application, it is shown that isomorphic spaces from this class have, up to quasidiagonal isomorphisms, the same basic subspaces of finite (infinite) type.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Isomorphisms of certain correlated probability spaces and of discrete memoryless correlated sources

Summary In an earlier paper [5], we defined a sufficient set of invariants for the isomorphy of discrete memoryless correlated sources with maximal correlation <1. Using the structure of isomorphisms of certain correlated probability spaces, we give here a sufficient set of invariants also for the case of maximal correlation equal to 1. We show, in particular, that two discrete memoryless stationary correlated sources with maximal correlation 1 may be isomorphic in a non-trivial way.     

The Moduli Space of n+1 Points in Complex Hyperbolic n-Space

The structure of the space of orbits of PU(n, 1) acting on (n+1)-tuples of points in complex hyperbolic space is characterized in terms of side lengths and angular invariants. The more general situation in which some of the points lie on the boundary of complex hyperbolic space is described in terms of other basic invariants.     

Endomorphisms of a Lebesgue space III

A new invariant is introduced for regular isomorphisms, which are isomorphisms by codes that anticipate a finite amount of the future. With the help of this invariant it is shown that the Bernoulli automorphism (p, q) is not regularly isomorphic to the Markov automorphism ( _pq ^qp ),p ≠q, and that neither of these is regularly isomorphic to the Markov automorphism ( _qp ^pq ).     

Geometric expansion,Lyapunov exponents and foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.     

Hermitian Morita Theory and Unitary Groups of Modules over Semisimple Artinian Rings

This article investigates isomorphisms between certain subgroups of the projective unitary groups of hermitian modules over semisimple Artinian rings with anti-structures. These subgroups contain the commutator subgroups of the projective unitary groups. Specifically, the article provides conditions under which these isomorphisms are induced by and the underlying rings are connected by hermitian Morita equivalences (HMEs). This article introduces the hyperbolic length of a module as well as the concept of generalized hyperbolic modules over simple Artinian rings, and over semisimple Artinian rings with anti-structures. The article shows that the stated isomorphisms are induced by HMEs if all the following conditions hold: (a) the hermitian forms are nonsingular and trace valued; (b) the modules in question are generalized hyperbolic; (c) the hyperbolic length equals three or is greater than or equal to five (hyperbolic length is greater than or equal to five in the semisimple case). Significantly, the condition of the hyperbolic length of a module greater than or equal to m is satisfied by a set of modules larger than or equal to those satisfying the condition of the Witt index of the module greater than or equal to m.     

Differential invariants for systems of linear hyperbolic equations

In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided.