Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

Free Jacobi Process

In this paper, we define and study two parameters dependent free processes (λ,θ) called free Jacobi, obtained as the limit of its matrix counterpart when the size of the matrix goes to infinity. The main result we derive is a free SDE analogous to that satisfied in the matrix setting, derived under injectivity assumptions. Once we did, we examine a particular case for which the spectral measure is explicit and does not depend on time (stationary). This allows us to determine easily the parameters range ensuring our injectivity requirements so that our result applies. Then, we show that under an additional condition of invertibility at time t=0, this range extends to the general setting. To proceed, we set a recurrence formula for the moments of the process via free stochastic calculus.     

Renyi entropy as a statistical entropy for complex systems

To describe a complex system, we propose using the Renyi entropy depending on the parameter q (0 < q ≤ 1) and passing into the Gibbs-Shannon entropy at q = 1. The maximum principle for the Renyi entropy yields a Renyi distribution that passes into the Gibbs canonical distribution at q = 1. The thermodynamic entropy of the complex system is defined as the Renyi entropy for the Renyi distribution. In contrast to the usual entropy based on the Gibbs-Shannon entropy, the Renyi entropy increases as the distribution deviates from the Gibbs distribution (the deviation is estimated by the parameter η = 1 − q) and reaches its maximum at the maximum possible value η_max. As this occurs, the Renyi distribution becomes a power-law distribution. The parameter η can be regarded as an order parameter. At η = 0, the derivative of the thermodynamic entropy with respect to η exhibits a jump, which indicates a kind of phase transition into a more ordered state. The evolution of the system toward further order in this phase state is accompanied by an entropy gain. This means that in accordance with the second law of thermodynamics, a natural evolution in the direction of self-organization is preferable. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 299–317, November, 2006.     

Invariant Pseudolinearity with Applications

In this paper, we introduce the notion of invariant pseudolinearity for nondifferentiable and nonconvex functions by means of Dini directional derivatives. We present some characterizations of invariant pseudolinear functions. Some characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, pseudolinear program are obtained. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs.     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

Invariant measures and disintegrations with applications to Palm and related kernels

Consider a locally compact group G acting measurably on some spaces S and T. We prove a general representation of G-invariant measures on S and the existence of invariant disintegrations of jointly invariant measures on S × T. The results are applied to Palm and related kernels associated with a stationary random pair (ξ,η), where ξ is a random measure on S and η is a random element in T. An erratum to this article can be found at     

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Strong entropy for system of isentropic gas dynamics

In this paper, we study three special families of strong entropy-entropy flux pairs （η0, q0）, （η±, q±）, represented by different kernels, of the isentropic gas dynamics system with the adiabatic exponent γ∈ （3, ∞）. Through the perturbation technique through the perturbation technique, we proved, we proved the H^-1 compactness of ηit ＋ qix, i = 1, 2, 3 with respect to the perturbation solutions given by the Cauchy problem （6） and （7）, where （ηi, qi） are suitable linear combinations of （η0, q0）, （η±, q±）.     

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.     

Orthogonal separable Hamiltonian systems on <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript>

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T ² for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T ² can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10671123, 10231020), “Dawn” Program of Shanghai Education Comission of China (Grant No. 03SG10) and Program for New Century Excellent Tatents in University of China (Grant No. 050391)     

Eta invariants for flat manifolds

Using a formula from Donnelly (Indiana Univ Math J 27(6):889–918, 1978), we prove that for a family of seven dimensional flat manifolds with cyclic holonomy groups the η invariant of the signature operator is an integer number. We also present an infinite family of flat manifolds with integral η invariant. The main motivation is a paper of Long and Reid (Geom Topol 4:171–178, 2000).     

Free products of groups have no outer normal automorphisms

An automorphism of an arbitrary group is called normal if all subgroups of this group are left invariant by it. Lubotski [1] and Lue [2] showed that every normal automorphism of a noncyclic free group is inner. Here we prove that every normal automorphism of a nontrivial free product of groups is inner as well. Supported by RFFR grant No. 13-011-1513. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 562–566, September–October, 1996.     

Entropy structure

Investigating the emergence of entropy on different scales, we propose an “entropy structure” as a kind of master invariant for the entropy theory of topological dynamical systems. An entropy structure is a sequence of functionsh _k on the simplex of invariant measures which converges to the entropy functionh and which falls into a distinguished equivalence class defined by a natural equivalence relation capturing the “type of nonuniformity in convergence”. An entropy structure recovers several existing invariants, including the symbolic extension entropy h_sex and the Misiurewicz parameter h^*. Entropy theories of Misiurewicz, Katok, Brin—Katok, Newhouse, Romagnoli, Ornstein—Weiss and others all yield candidate sequences (h _k); we determine which of these exhibit the correct type of convergence and hence become entropy structures. One of the satisfactory sequences arises from a new treatment of entropy theory strictly in terms of continuous functions (in place of partitions or covers). The results allow the computation of symbolic extension entropy without reference to zero dimensional extensions. New light is shed on the property of asymptotich-expansiveness. Supported by the KBN grant 2 P03 A 04622.     

Reflecting topological properties in continuous images

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ ⁺ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ ⁺. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ ⁺ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ ⁺ for compact spaces.     

Regularization by Free Additive Convolution, Square and Rectangular Cases

The free convolution \boxplus\boxplus is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution \boxplus_l\boxplus_{\lambda} allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups (μ_t) of probability measures such that μ₀ = δ₀ and such that for all t > 0 and all probability measure n, m_t\boxplusn\nu, \mu_t\boxplus\nu (or, in the rectangular context, m_t\boxplus_ln\mu_t\boxplus_{\lambda}\nu) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for \boxplus\boxplus, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for μ in a \boxplus_l\boxplus_{\lambda}-continuous semigroup, m\boxplus_ln\mu\boxplus_{\lambda}\nu either has an atom at the origin or doesn’t put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of m\boxplus_ln\mu\boxplus_{\lambda}\nu except on a negligible set of points, as well as existence and continuity of a density everywhere.     

Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version of Stokes’ theorem, and establish a bounded tangential version of Poincaré’s Lemma. These results are made possible by the structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer [Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincaré’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology. Supported by BSF and ISF. Supported by BSF and NSF.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

Free entropy dimension in amalgamated free products

We calculate the microstates free entropy dimension of naturalgenerators in an amalgamated free product of certain von Neumannalgebras, with amalgamation over a hyperfinite subalgebra. Inparticular, some ‘exotic’ Popa algebra generatorsof free group factors are shown to have the expected free entropydimension. We also show that microstates and non-microstatesfree entropy dimension agree for generating sets of many groups.In the appendix, the first L²-Betti number for certain amalgamatedfree products of groups is calculated.     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Smooth interval maps have symbolic extensions

We prove that if X denotes the interval or the circle then every transformation T:X→X of class C ^r , where r>1 is not necessarily an integer, admits a symbolic extension, i.e., every such transformation is a topological factor of a subshift over a finite alphabet. This is done using the theory of entropy structure. For such transformations we control the entropy structure by providing an upper bound, in terms of Lyapunov exponents, of local entropy in the sense of Newhouse of an ergodic measure ν near an invariant measure μ (the antarctic theorem). This bound allows us to estimate the so-called symbolic extension entropy function on invariant measures (the main theorem), and as a consequence, to estimate the topological symbolic extension entropy; i.e., a number such that there exists a symbolic extension with topological entropy arbitrarily close to that number. This last estimate coincides, in dimension 1, with a conjecture stated by Downarowicz and Newhouse [13, Conjecture 1.2]. The passage from the antarctic theorem to the main theorem is applicable to any topological dynamical system, not only to smooth interval or circle maps.