Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:?• The uniform norm of the differential of its n-th iteration;?• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.?We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups. Oblatum 6-XII-2001 & 19-VI-2002?Published online: 5 September 2002 RID="*" ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Separating diffeomorphisms of the torus

There exists a diffeomorphism on the n-dimensional torus Tⁿ which is conjugate with a hyperbolic linear automorphism, but is not an Anosov diffeomorphism. A diffeomorphismf: Tⁿ→Tⁿ has such a property iff is separating and belongs to the C⁰ closure of the Anosov diffeomorphisms.     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday     

Dehn twists and pseudo-Anosov diffeomorphisms

Summary We show that it is possible to obtain many pseudo-Anosov diffeomorphisms from Dehn twists. In particular, we generalize a theorem of Long and Morton to obtain that iff is a pseudo-Anosov diffeomorphism of an oriented surface andT is the Dehn twist around the simple closed curve , then the isotopy class ofT ⁿ f contains a pseudo-Anosov diffeomorphism except for at most 7 consecutive values ofn.     

More Denjoy minimal sets for area preserving diffeomorphisms

For an area preserving, monotone twist diffeomorphism and an irrational number ω, we prove that if there is no invariant circle of angular rotation number ω, then there are uncountably many Denjoy minimal sets of angular rotation number ω. For each pair of positive integersn andR we prove that the space (with the vague topology) of Denjoy minimal sets of angular rotation number ω and intrinsic rotation number (ω+R)/n (mod. 1) contains a disk of dimensionn−1. Partially supported by NSF contract #MCS82-01604.     

Espace des modules de certains polyèdres projectifs miroirs

A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in (0,\fracp2]{(0,\frac{\pi}{2}]}, and the union of n copies of \mathbbR^d{\mathbb{R}^{d}}, when the polyhedron has the combinatorics of an ecimahedron, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers n and d can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

Diff(Rn) as a Milnor–Lie group

We describe a construction of the Lie group structure on the diffeomorphism group Diff( R ⁿ), modelled on the space D( R ⁿ, R ⁿ) of R ⁿ‐valued test functions on R ⁿ, in John Milnor's setting of infinite‐dimensional Lie groups. New tools are introduced to simplify this task. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Représentations du groupe des tresses généralisées dans des groupes de Lie

Let Γ be a Lie group.Then any automorphism of the free group of rank n induces a diffeomorphism of Γ ⁿ . We use this remark and a result of P. Vogel to construct linear representations of a certain automorphism group of the algebraic closure of the free group. This automorphism group is closely related to the string link cobordism group.

Received:     

Flat Manifolds With Prescribed First Betti Number Admitting Anosov Diffeomorphisms

 We show that from dimension six onwards (but not in lower dimensions), there are in each dimension flat manifolds with first Betti number equal to zero admitting Anosov diffeomorphisms. On the other hand, it is known that no flat manifolds with first Betti number equal to one support Anosov diffeomorphisms. For each integer k > 1 however, we prove that there is an n-dimensional flat manifold M with first Betti number equal to k carrying an Anosov diffeomorphism if and only if M is a k-torus or n is greater than or equal to k + 2. (Received 5 October 2000; in revised form 9 March 2001)     

Compositions with distinct parts

The number of compositionsC(n) of a positive integern into distinct parts can be considered as a natural analogue of the numberq(n) of distinct partitions ofn. We obtain an asymptotic estimate forC(n) and in addition show that the sequence {C(n, k)} of distinct compositions ofn withk distinct parts is unimodal. Our analysis is more complicated than is usual for composition problems. The results imply however that the behaviour of these functions is of comparable complexity to partition problems.     

Two Characterizations of Diffeomorphisms of Euclidean Space

Two characterizations for a local diffeomorphism of R^n to be global one are given in terms of associated Wazewski equations.The two characterizations could be useful for the investigation of the Jacobian conjecture.     

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F _n , , defined in L ²(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F _n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F _n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.     

On the simple isotopy class of a source-sink diffeomorphism on the 3-sphere

The results obtained in this paper are related to the Palis-Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse-Smale systems on a closed smooth manifold M ⁿ . Newhouse and Peixoto showed that such an arc joining flows exists for any n and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For n = 1, this is related to the presence of the Poincaré rotation number, and for n = 2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension n = 3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse-Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces on domains

The purpose of this paper is to develop a theory of the Besov‐Morrey spaces and the Triebel‐Lizorkin‐Morrey spaces on domains in R ⁿ. We consider the pointwise multiplier operator, the trace operator, the extension operator and the diffeomorphism operator. Not only to domains in R ⁿ we extend our definition of function spaces to compact oriented Riemannian manifolds. Among the properties above, the result for the trace operator is in particular interesting, which reflects the property of the parameters p, q in the Morrey space ??^p_q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Identities of unitary finite-dimensional algebras

We deal with growth functions of sequences of codimensions of identities in finite-dimensional algebras with unity over a field of characteristic zero. For three-dimensional algebras, it is proved that the codimension sequence grows asymptotically as a ⁿ , where a is 1, 2, or 3. For arbitrary finite-dimensional algebras, it is shown that the codimension growth either is polynomial or is not slower than 2 ⁿ . We give an example of a finite-dimensional algebra with growth rate an with fractional exponent a = \frac3³?{4} + 1 a = \frac{3}{{\sqrt[3]{4}}} + 1 .     

Equidistribution of sparse sequences on nilmanifolds

We study equidistribution properties of nil-orbits (b ⁿ x) _n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ ℝ \ ℤ is positive, then the sequence $ (b^{[n^c ]} x)_{n \in \mathbb{N}} $ (b^{[n^c ]} x)_{n \in \mathbb{N}} is equidistributed in X for every x ∈ X. This is also the case when n ^c is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when X is the circle.     

Constructing infra-nilmanifolds admitting an Anosov diffeomorphism

In this paper we establish an algebraic characterization of those infra-nilmanifolds modeled on a free c-step nilpotent Lie group and with an abelian holonomy group admitting an Anosov diffeomorphism. We also develop a new method for constructing examples of infra-nilmanifolds having an Anosov diffeomorphism.