Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on \mathbb Rⁿ{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the case of random variables in \mathbb Rⁿ{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well.     

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.     

关于连续映射半群拓扑熵的一点注记

本文研究了度量空间中连续映射构成半群的拓扑熵.利用Patr′ao~([8])的方法,给出了度量空间中两种有限个连续映射构成的半群的拓扑d-熵的定义,比较了两种拓扑d-熵的大小.证明了局部紧致可分度量空间上有限个真映射构成的半群的拓扑d-熵和它的一点紧化空间上对应的拓扑熵相等.上面结果推广了Patr′ao的相应结论.     

Integrable Hamiltonian systems with positive topological entropy

In this paper we provide a class of integrable Hamiltonian systems on a three-dimensional Riemannian manifold whose flows have a positive topological entropy on almost all compact energy surfaces. As our knowledge, these are the first examples of C^∞ Liouvillian integrable Hamiltonian flows with potential energy on a Riemannian manifold which has a positive topological entropy.     

Differentiability and analyticity of topological entropy for Anosov and geodesic flows

Summary In this paper we investigate the regularity of the topological entropyh _top forC ^k perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].Partially supported by NSF Grant DMS85-14630     

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ^∗-algebra.     

Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

Eine bemerkung zur topologischen entropie

We study fractional quadratic transformationsT of the sphere and try to determine their topological entropy. In the case whereT is a constant mapping or a homeomorphism, the topological entropy is of course zero. In the other cases, we have the following results. IfT has only one fixed point, its entropy is log 2. IfT has exactly two fixed points, it can be written asT _z =z–z ^–1 +v, and ifv is real, then the entropy ofT is again log 2. A general result ofMisiurewicz andPrzytycki shows that the entropy ofT is at least log2, and we conjecture that this entropy is always equal to log2 in the remaining cases, i. e. two fixed points andv not real, and three fixed points.     

A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk

We consider homeomorphisms ƒ of a punctured 2-disk D ² \ P, where P is a finite set of interior points of D ², which leave the boundary points fixed. Any such homeomorphism induces an automorphism ƒ _* of the fundamental group of D ² \ P. Moreover, to each homeomorphism ƒ, a matrix B _ƒ (t) from GL(n, ℤ[t, t ⁻¹]) can be assigned by using the well-known Burau representation. The purpose of this paper is to find a nontrivial lower bound for the topological entropy of ƒ. First, we consider the lower bound for the entropy found by R. Bowen by using the growth rate of the induced automorphism ƒ _*. Then we analyze the argument of B. Kolev, who obtained a lower bound for the topological entropy by using the spectral radius of the matrix B _ƒ (t), where t ∈ ℂ, and slightly improve his result. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 47–55, 2005.     

Random C 0 homeomorphism perturbations of hyperbolic sets

In this note we consider random C ⁰ homeomorphism perturbations of a hyperbolic set of a C ¹ diffeomorphism. We show that the hyperbolic set is semi-stable under such perturbations, in particular, the topological entropy will not decrease under such perturbations.     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L ¹ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ^∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.     

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g ^-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank _h X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank _h conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H ⁿ with . Submitted: February 2001, Revised: October 2001.     

Algebraic and topological entropy on lie groups

This paper starts with some examples and quick results on the topological entropy of continuous functions. It discusses the topological entropy on Lie groups and proves their shift properties. It proves Fried's conjecture h(φγ) <- h(φ)+h(γ) for affine maps on Lie groups. Moreover, φ and γ do not have to commute. As a corollary, it proves that entropy is invariant with isometric endomorphisms of Lie groups. Also, it discusses algebraic entropy on elementary Abelian groups and Lie groups. It proves that the topological entropy is preserved when projected from Lie group lib to its quotient space compact Lie group ^S1 for continuous functions lifted from the quotient space and shows that algebraic entropy in general is strictly less than topological entropy.     

Real analytic convex surfaces with positive topological entropy and rigid body dynamics

Following Knieper and Weiss [9] we exhibit explicit real analytic metrics onS ² andR P ² with positive curvature and positive topological entropy using the dynamics of the rigid body. Supported by the Max-Planck-Institut für Mathematik and by a travel grant from CDE.     

The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one-dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C¹ map on any compact manifold.     

Amenable groups, topological entropy and Betti numbers

We investigate an analogue of theL ²-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship.