Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric \(D_B\) for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to \(D_B\). It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure \(\mu \) of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for \(\mu \). A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.     

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests.     

The weak specification property and distributional chaos

In this paper, we first discuss almost periodic points in a compact dynamical system with the weak specification property. On the basis of this discussion, we draw two conclusions: (i) the weak specification property implies a dense Mycielski uniform distributionally scrambled set; (ii) the weak specification property and a fixed point imply a dense Mycielski uniform invariant distributionally scrambled set. These conclusions improve on some of the latest results concerning the specification property, and give a final positive answer to an open problem posed in [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), 31–43].     

几乎周期性与SS混沌集

引进正则移位不变集的概念,证明了有正则移位不变集的紧致系统在几乎周期点集中存在SS混沌集,特别地,具有正拓扑熵的区间映射在几乎周期点集中存在SS混沌集.     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

On the mean square of the remainder for the euclidean lattice point counting problem

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.     

On the absence of invariant measures with locally maximal entropy for a class of shifts of finite type

We prove that for a class of shifts of finite type, , any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.

     

Some Weak Specification Properties and Strongly Mixing

In this paper,the authors first construct a dynamical system which is strongly mixing but has no weak specification property.Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper,respectively,and the authors prove the equivalence of quasi-weak specification property,semi-weak specification property and strongly mixing.     

On Quasi-weakly Almost Periodic Points of Continuous Flows

Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.     

Shadowing, entropy and minimal subsystems

We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.     

Tail invariant measures of the Dyck shift

We show that the one-sided Dyck shift has a unique tail invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. For the two sided Dyck shift we show that there are exactly three ergodic double-tail invariant probabilities. We show that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy. This article is a part of the author’s M.Sc. Thesis, written under the supervision of J. Aaronson, Tel-Aviv University.     

Two New Recurrent Levels for <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-Flows

The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ω-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.     

Embedding asymptotically expansive systems

A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if \( h_{top}(T)<\log K\) and \(\sharp Per_n(X,T)\le K^n\) for any integer n. The embedding is in general not continuous (unless the system is expansive and X is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension.     

SHADOWING,EXPANSIVENESS AND SPECIFICATION FOR C~1-CONSERVATIVE SYSTEMS

We prove that a C~1-generic volume-preserving dynamical system(diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov.Finally,as in[10,27],we prove that the C~1-robustness,within the volume-preserving context,of the expansiveness property and the weak specification property,imply that the dynamical system(diffeomorphism or flow) is Anosov.     

Expanding measures

We prove that any C1+α transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller (1990) [33] for maps of the interval with negative Schwarzian derivative.Given a non-uniformly expanding set, we also show how to construct a Markov structure such that any invariant measure defined on this set can be lifted. We used these structure to study decay of correlations and others statistical properties for general expanding measures.     

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.     

Markov shift in non-commutative probability

We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of ‘quantum detailed balance’ introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.     

Minimal self-joinings and positive topological entropy

We show that the properties of almost minimal self-joinings and strong almost minimal self-joinings, introduced by del Junco in Topological Dynamics, are compatible with positive topological entropy, as opposed to the stronger property of minimal self-joinings. This is done both by proving existence theorems and by explicitly constructing some symbolic systems having these properties, which are modifications of the Chacón system. It is shown furthermore that these systems have no non-trivial factors with completely positive topological entropy.     

Invariant measure and Lyapunov exponents for birational maps of P2

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.