Smooth Conjugacy Research Articles

Golden tilings

We introduce the notion of golden tilings, and we prove a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, (ii) affine classes of golden tilings and (iii) solenoid functions. The solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences.

Nonhomogeneous locally free actions of the affine group

We classify locally free smooth actions of the real affine group on closed three-dimensional manifolds up to smooth conjugacy. In particular, we show that there are non-homogeneous actions when the manifold is not a rational homology sphere and its fundamental group is not solvable. We say two right actions ρ1 and ρ2 of a Lie group G on manifolds M1 and M2 are C ∞ conjugate if there exist an isomorphism σ of G and a C ∞ diffeomorphism H from M1 to M2 such that H(ρ1(p, g)) = ρ2(H(p), σ(g)) for any g 2 G and p 2 M1. The map H is called a C ∞ conjugacy between ρ1 and ρ2. We also say ρ1 and ρ2 are C ∞ orbit equivalent if there exists a C ∞

On smooth conjugacy of expanding maps in higher dimensions

In this paper we investigate smooth conjugacy of $C^\infty$ expanding maps on certain nilmanifolds. We show that several rigidity results about expanding maps on the circle can not be generalized directly to higher dimensions. For example the following result is obtained: Let $\Gamma_1\backslash N_1$ and $\Gamma_2\backslash N_2$ be two nilmanifolds of homogeneous type. We show that for any positive integer $k$ there exist on the product nilmanifold $\Gamma_1\times \Gamma_2\backslash N_1\times N_2$ a $C^\infty$ expanding map $\varphi$ and an expanding nilendomorphism $\psi$ which are $C^k$ conjugate, but not $C^{k,lip}$ conjugate. While in the case of dimension one, it was shown by M. Shub and D. Sullivan that if two $C^\infty$ expanding maps on $\mathbb S^1$ are absolutely continuously conjugate, then they must be $C^\infty$ conjugate.

Smooth conjugacy of circle diffeomorphisms with break

We prove that every two circle diffeomorphisms with break that have the same size of break (i.e., the ratio of the left and right derivatives at the break point) and the same irrational revolution number from a certain class that contains non-Diophantine numbers are C 1-smoothly conjugate.

Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+\nu}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same constant periodic eigenvalue data that are only Holder conjugate. We show that these examples exhaust all possible ways to perturb a $C^{1+\nu}$ conjugacy class without changing any periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to the original example.

Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

We consider germs of conformal mappings tangent to the identity at the origin in C. We construct a germ of a homeomorphism which is a C ∞ diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field V(z) = z m ∂ ∂z. We then show that, in the case m = 2, for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.

Geometry of KAM tori for nearly integrable Hamiltonian systems

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

Unicity of KAM tori

The classical KAM theorem establishes the persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. A long-standing question is to what extent the perturbed tori are unique. Using the fact that at the level of tori there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine torus union of full measure.

Smooth conjugacy between S-unimodal maps

It is proved that if two Cr (r ≥ 3) S-unimodal maps without periodic or Cantor attractors are conjugate, and the conjugacy preserves the multipliers of periodic points, then the conjugacy is Cr. This solves a conjecture of Martens and de Melo.

Sternberg theorems for random dynamical systems

AbstractIn this paper, we prove the smooth conjugacy theorems of Sternberg type for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy. © 2005 Wiley Periodicals, Inc.

Stable manifolds and the Perron–Irwin method

We establish rather general and simple theorems implying, among other things, the pseudo-(un)stable manifold theorem, Sternberg's theorem on smooth conjugacy between hyperbolic germs of maps or vector fields, and results of Fenichel, Hirsch, Pugh and Shub on existence, uniqueness and structural stability of stable or unstable manifolds at compact invariant manifolds. The Perron–Irwin approach via sequence spaces plays a crucial role.

A class of generalized symmetries of smooth flows

A class of generalized space-time symmetries is defined by extending the notions of classical symmetries and reversing symmetries for a smooth flow to arbitrary constant reparameterizations of time. This class is shown to be the group-theoretic normalizer of the abelian group of diffeomorphisms generated by the flow. Also, when the flow is nontrivial, this class is shown to be a nontrivial subgroup of the group of diffeomorphisms of the manifold, and to have a one-dimensional linear representation in which the image of a generalized symmetry is its unique constant reparameterization of time. This group of generalized symmetries and several groups derived from it (among which are the multiplier group and the reversing symmetry group) are shown to be nontrivial but incomplete invariants of the smooth conjugacy class of a smooth flow. Several examples are given throughout to illustrate the theory.

Equivariant Sternberg–Chen Theorem

Smooth (C∞) vector fields with linear symmetries and anti-symmetries are considered. We prove that provided symmetry group is compact then a smooth conjugacy in Sternberg–Chen Theorem can be chosen symmetric.

Some results on stable manifolds

We state very general and simple non-perturbative theorems and sketch part of their proof. They imply, among other things [2], results of Fenichel [3], Hirsch, Pugh and Shub [4] on stable or unstable manifolds at compact invariant manifolds, the pseudo-(un)stable manifold theorem at a fixed point [4,6,7] and Sternberg's theorem [8,9] on smooth conjugacy of hyperbolic gems of maps or vector fields. The Perron–Irwin approach via sequence spaces [5,6,7] plays a crucial role in the proofs.

Trace formulae for stochastic evolution operators: smooth conjugation method

The trace formula for the evolution operator associated with nonlinear stochastic flows with weak additive noise is cast in the path integral formalism. We integrate over the neighbourhood of a given saddlepoint exactly by means of a smooth conjugacy, a locally analytic nonlinear change of field variables. The perturbative corrections are transferred to the corresponding Jacobian, which we expand in terms of the conjugating function, rather than the action used in defining the path integral. The new perturbative expansion which follows by a recursive evaluation of derivatives appears more compact than the standard Feynman diagram perturbation theory. The result is a stochastic analogue of the Gutzwiller trace formula with the `' corrections computed an order higher than what has so far been attainable in stochastic and quantum mechanical applications.

The multipliers of periodic points in one-dimensional dynamics

It will be shown that the smooth conjugacy class of an S-unimodal map which has neither a periodic attractor nor a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M Shub and D Sullivan (1985 Expanding endomorphism of the circle revisited Ergod. Theor. Dynam. Sys. 5 285-9) for smooth expanding maps of the circle.

Relative density of irrational rotation numbers in families of circle diffeomorphisms

Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

Classifying C1+ structures on dynamical fractals: 2. Embedded trees

AbstractWe classify the C1+α structures on embedded trees. This extends the results of Sullivan on embeddings of the binary tree to trees with arbitrary topology and to embeddings without bounded geometry and with contact points. We used these results in an earlier paper to describe the moduli spaces of smooth conjugacy classes of expanding maps and Markov maps on train tracks. In later papers we will use those results to do the same for pseudo-Anosov diffeomorphisms of surfaces. These results are also used in the classification of renormalisation limits of C1+α diffeomorphisms of the circle.

Classifying C1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks

AbstractSullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d − 1}ℕ satisfying the matching condition.

Renormalisation of bicritical circle maps

For invertible maps of the circle with two cubic critical points, called “bicritical”, an additional invariant to the rotation number is identified, under smooth conjugacy. The way it evolves under renormalisation is derived. Conjectures are proposed about the dynamics of renormalisation on the space of bicritical circle maps.