Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.     

Bernoulli shifts and local density property

We consider the property LocDen for the squaring mapping into the space of all measure preserving transformations and into the space of mixing transformations. It is proved that Bernoulli shifts with infinite entropy do not possess this property.     

Pre-image entropy for free semigroup actions

The purpose of this study is to indicate fundamental propositions of the pre-image entropy related to a system proposed by Bufetov [Bufetov A. Topological entropy of free semigroup actions and skew-product transformations. J Dyn Control Syst 1999;5:137–143. 1.] for generating free semigroup actions. This study reveals the formula for the pre-image entropy of skew-product transformation with respect to the one-sided shift space. Finally, one example is presented to show how to obtain the pre-image entropy value for the skew-product transformation.     

On marginal distributions and isomorphisms of stationary processes

LetT be an invertible ergodic aperiodic measure preserving transformation of a Lebesgue space, letA be a finite alphabet, and let π be a probability measure onA ⁿ which admits a mixing shift-invariant measureμ _π onΩ=A ^? such that the marginals of anyn successive coordinates are π and the entropyh(T) ofT is smaller than the entropy of the shift in (Ω,μ _π). Then there exists a shift invariant measure ν_π in Ω which also has marginals π and for whichT is isomorphic to the shift in (Ω, ν_π). This contains Krieger's finite generator theorem and strengthens the measure theoretic part of his approximation theorem for shift-invariant measures by showing that the preassigned marginal π can not only be achieved up to an ε>0 but exactly. Our result also contains an as yet unpublished theorem of Krieger, which says thatT can be embedded in an arbitrary mixing subshift of finite type, as long as the entropy of the subshift under the measure with maximal entropy exceeds that ofT. In the final section we show that the method can be extended to yield also exact marginals for the generator in the Jewett-Krieger theorem, i.e.T is shown to be isomorphic to a shift in (Ω, ν_π) where ν_π has exact marginals π and the shift is uniquely ergodic on the support of ν_π.     

Dynamique des automorphismes des surfaces projectives complexes

We study the dynamics of automorphisms of complex projective surfaces. Letbe such an automorphism whose topological entropy is not zero. We construct a probability measure associated toand the complex structure. This measure is -invariant, ergodic and has maximal entropy. This is the unique measure satisfying these properties and periodic points are equidistributed with respect to this measure.     

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.     

One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II₁ factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II₁ factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.     

Invariant gauge automorphisms of homogeneous principal bundles

The inner and outer automorphism groups of a Lie group are generalized by considering automorphisms in the category of homogeneous principal bundles. These automorphisms are then used to produce certain invariant gauge transformations of such bundles. Some aspects of the resulting action on the space of invariant connections are also described.     

Dynamics measured in a non-Archimedean field

We study dynamical systems using measures taking values in a non-Archimedean field. The underlying space for such measure is a zero-dimensional topological space. In this paper we elaborate on the natural translation of several notions, e.g., probability measures, isomorphic transformations, entropy, from classical dynamical systems to a non-Archimedean setting.     

Ergodicity of Rokhlin cocycles

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions. We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations. Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems. Research partially supported by KBN grant 2 P03A 002 14 (1998).     

Families of type III0 ergodic transformations in distinct orbit equivalent classes

A new isomorphism invariant of certain measure preserving flows, using sequences of integers, is introduced. Using this invariant, we are able to construct large families of type III₀ systems which are not orbit equivalent. In particular we construct an uncountable family of nonsingular ergodic transformations, each having an associated flow that is approximately transitive (and therefore of zero entropy), with the property that the transformations are pairwise not orbit equivalent.     

Families of type III<Subscript>0</Subscript> ergodic transformations in distinct orbit equivalent classes

A new isomorphism invariant of certain measure preserving flows, using sequences of integers, is introduced. Using this invariant, we are able to construct large families of type III₀ systems which are not orbit equivalent. In particular we construct an uncountable family of nonsingular ergodic transformations, each having an associated flow that is approximately transitive (and therefore of zero entropy), with the property that the transformations are pairwise not orbit equivalent.     

Applying projective geometry to transformations on rank one idempotents

We present a short proof of Molnár's characterization of bijective transformations on the set of all rank one idempotent operators on a Banach space which preserve zero products in both directions. An improvement in the finite-dimensional case is given. We apply these results to describe automorphisms of standard operator semigroups and to improve Uhlhorn's version of Wigner's theorem.     

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.     

DISTINGUISHED LINE BUNDLES FOR COMPLEX SURFACE AUTOMORPHISMS

We equate dynamical properties (e.g., positive entropy, existence of a periodic curve) of complex projective surface automorphisms with properties of the pull-back actions of such automorphisms on line bundles. We use the properties of the cohomological actions to describe the measures of maximal entropy for automorphisms with positive entropy.     

The structure of skew products with ergodic group automorphisms

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.     

Pseudodifferential <Emphasis Type="Italic">p</Emphasis>-adic vector fields and pseudodifferentiation of a composite <Emphasis Type="Italic">p</Emphasis>-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O _p ) of balls in O _p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O _p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.