Herman’s theory revisited

We prove that a C ^2+α -smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D _δ , 0≤δ<α≤1, α−δ≠1, is C ^1+α−δ -smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms.     

Definition of Measure-theoretic Pressure Using Spanning Sets

We introduce a new definition of measure–theoretic pressure for ergodic measures of continuous maps on a compact metric space. This definition is similar to those of topological pressure involving spanning sets. As an application, for C ^{1+ α} (α > 0) diffeomorphisms of a compact manifold, we study the relationship between the measure–theoretic pressure and the periodic points. Project Supported by National Natural Science Foundation of China     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Reducing conjugacy in the full diffeomorphism group of ℝ to conjugacy in the subgroup of orientation-preserving maps

Let Diffeo = Diffeo(ℝ) denote the group of infinitely differentiable diffeomorphisms of the real line ℝ, under the operation of composition, and let Diffeo⁺ be the subgroup of diffeomorphisms of degree +1, i.e., orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo⁺. The main result concerns the case when f and g have degree −1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo⁺, in order to ensure that f is conjugated to g by an element of Diffeo⁺. The methods involve formal power series and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibliography: 4 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 231–237.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in on ∂Ω, with α a fixed real, and a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable. Robert Smits: This author was partially supported by a grant of the National Security Agency, grant #H98230-05-1-0060.     

On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product

Regularity for minimal boundaries in R n ¶with mean curvature in L n

Let be a minimal set with mean curvature in L ⁿ that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C ^α mapping with C ^α inverse. Received: 11 July 1997 / Revised version: 24 February 1998     

Examples of C1-smoothly conjugate diffeomorphisms of the circle with break that are not C1+γ-smoothly conjugate

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C ^1+γ -smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C ¹-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.     

Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle

We prove that stable ergodicity is C ^r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞]. The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C ^r open and dense, and essential accessibility implies ergodicity. Mathematics Subject Classification (2000) Primary: 37D30, Secondary: 37A25     

Regular interval Cantor sets of <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> and minimality

It is known that not every Cantor set of S ¹ is C ¹-minimal. In this work we prove that every member of a subfamily of what we here call regular interval Cantor set is not C ¹-minimal. We also prove that no member of a class of Cantor sets that includes this subfamily is C ^1+∈-minimal, for any ∈ > 0. Partially supported by CNPq-Brasil and PEDECIBA-Uruguay.     

Some inverse<Emphasis Type="Italic">M</Emphasis>-matrix problems

An upper bound and a lower bound for α⁰ are given such that for for α≤α⁰, whereB is a nonnegative matrix and satisfies that for any positive constant β,βI+B is a power invariant zero pattern matrix. This project is supported by Science and Art Foundation of Central South University of Technology.     

The Adjoint Action of an Expansive Algebraic -Action

 Let , and let α be an expansive -action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup is a -action by automorphisms of . By duality, there exists a -action by automorphisms of the compact abelian group : this action is called the adjoint action of α. We prove that is again expansive and has completely positive entropy, and that α and are weakly algebraically equivalent, i.e. algebraic factors of each other. A -action α by automorphisms of a compact abelian group X is reflexive if the -action on the compact abelian group adjoint to is algebraically conjugate to α. We give an example of a non-reflexive expansive -action α with completely positive entropy, but prove that the third adjoint is always algebraically conjugate to . Furthermore, every expansive and ergodic -action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic -actions with zero entropy. Received 11 June 2001; in revised form 29 November 2001     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .