Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

Shift Automorphisms of the Hyperfinite Factor

For a large class of shift transformations of a LEBESGUE measure space (they have to fulfil some mixing condition) we construct automorphisms of the hyperfinite factor of type II₁. The CONNES -STØRMER entropy of the resulting automorphisms equals the measure theoretic entropy of the corresponding shift transformations. Two such automorphisms are conjugate if the conjugacy between the original measure space shifts can be given by a code with finite expected code length.     

Equivalence of Differential System

Using reflecting function of Mironenko we construct some differential systems which are equivalentto the given differential system.This gives us an opportunity to find out the monodromic matrix of these periodicsystems which are not integrable in finite terms.     

On the piecewise smoothness of entropy solutions to scalar conservation laws

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C¹-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C¹-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cⁿ-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C¹-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.     

On phase transitions for subshifts of finite type

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.     

Entropy in the category of perfect complexes with cohomology of finite length

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic.Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is “additive”. Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.     

Some characterization results on generalized cumulative residual entropy measure

The cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon entropy, refer to Rao et al. (2004). In this paper we study a generalized cumulative residual information measure based on Verma’s entropy function and a dynamic version of it. The exponential, Pareto and finite range distributions, which are commonly used in reliability modeling, have been characterized using this generalized measure.     

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.     

Finite entropy actions of free groups,rigidity of stabilizers,and a Howe-Moore type phenomenon

We study a notion of entropy, called f-invariant entropy, introduced by Lewis Bowen for probability measure preserving actions of finitely generated free groups. In the degenerate case, the f-invariant entropy is -∞. In this paper, we investigate the qualitative consequences of an action having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov-Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe-Moore property. Specifically, if the action is ergodic, there exists an integer n such that for every non-trivial normal subgroup K, the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.     

幂零流形上自映射的点态原像熵的可加性

类似于拓扑熵,点态原像熵作为动力系统的不变量,也度量了紧度量空间上系统的复杂性.但至今不知其性质与拓扑熵是否完全一致,例如映射笛卡尔积的点态原像熵的可加性等.本文将把环面自映射笛卡尔积的点态原像熵的可加性,推广到紧幂零流形自映射的情形.     

三叉树模型下的等价鞅测度

本文研究了三叉树模型下的等价鞅测度刻划问题，得到了三叉树模型的最小熵鞅测度，逆相对熵鞅测度，方差最优鞅测度和极小鞅测度的精确表达式。     

CONVERGENCE OF AN EXPLICIT UPWIND FINITE ELEMENT METHOD TO MULTI-DIMENSIONAL CONSERVATION LAWS

1. IntroductionThe convergence problem for the numerical schemes to one dimensional cons~inn lawshas been edensively studied. By tensor product one dimensional schemes can be appliedto multi-dimensional equations. However the convergence of many of those schemes is stillchanknown even if it is true for one dimensional cases. Besides, for those Phys.ical domains withcomplicated geometry unstructured grids are more fiekible. In recent yeaes the convergellceproblem for unstructured grids has cal…     

On the entropy of a product endomorphism in infinite measure spaces

Summary In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a -finite measure space, and the notion of a spreading partition. This relationship is used to answer in the quasi-finite case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and -finite measure space.     

Liouville property for groups and manifolds

We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy. Mathematics Subject Classification (1991) 60B15, 60J50, 28D20, 20P05, 43A07, 60J65, 43A85, 20f16     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

具有零拓扑熵的图映射的攀援集的测度

本文研究了具有零拓扑熵的图映射f的性质,证明了在任意有限f-不变的Borel 测度μ下,其攀援集的外μ-测度都是零.     

Entropy of conservative transformations

Summary The definition of entropy of a measure-preserving transformation (called: endomorphism) of a finite measure space into itself makes no sense for -finite measure spaces. Using induced transformations (introduced by Kakutani [1]) we give a definition which applies to conservative endomorphisms in -finite measure spaces. (This covers all cases of interest, since dissipative endomorphisms have a rather simple structure.) A theorem of Abramov [2] implies that for finite measure spaces the new definition is equivalent to the old one. Entropy as a metric invariant of conservative transformations has many, but not all of the properties discovered by Kolmogorov, Sinai, Rokhlin and others in the finite case. Major differences between the finite and the -finite case occur in the investigation of transformations with entropy 0.After giving the basic definitions in section 1 we first prove a theorem on antiperiodic transformations, which will be needed in all other sections, unless the reader is willing to assume that all transformations are ergodic. In section 3 we define entropy and prove a theorem which permits its computation. As an example the entropy of the Markov shift for null-recurrent Markov chains is computed in section 4. We then investigate simple properties such as h(T ⁿ )=nh(T) (section 5) and give the ergodic decomposition of h(T) in section 6. Section 7 is devoted to the investigation of transformations with entropy zero, especially an example is given which shows that a known necessary and sufficient condition for a transformation with finite invariant measure to have entropy zero is not sufficient for transformations with a -finite invariant measure unless they satisfy an additional assumption. Finally section 8 is devoted to the proof of category statements about the set of conservative transformations and the subset of those among them which have entropy zero.Prepared with the partial support of the National Science Foundation, Grant. No. GP-2593.Die übersetzung der vorliegenden Arbeit ins Deutsche wurde von der Naturwissenschaftlichen FakultÄt der Friedrich-Alexander-UniversitÄt Erlangen-Nürnberg im WS 1966/67 als Habilitationsschrift angenommen.I would like to thank Mr. H. Scheller for providing me with a copy of his unpublished paper [9]. My thanks are also due to Professor K. Jacobs, whose lectures made me familiar with the theory generalized in this paper and who kept me informed about some recent results.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer’s utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.     

Ergodic automorphisms of the infinite torus are bernoulli

We show that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts. Using the same techniques, we also show that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.