Smooth Conjugacy Research Articles

Anosov endomorphisms on the two-torus: regularity of foliations and rigidity

We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the two-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterisation of smooth conjugacy between special Anosov endomorphisms (defined as those having only one unstable direction for each point) and their linearisations.

A Note on Equivalence Classes of SO0(p,q)-actions on Sp+q-1 whose Restricted SO(p)×SO(q)-action is the Standard Action

In [5] we gave examples of triples (g,h1,h2) of smooth functions to construct smooth SU(p,q)-actions on the projective space Pℂp+q-1 (p,q≥3) whose restricted S(U(p)×U(q))-action is the standard action. By using these examples of triples (g,h1,h2), we shall construct smooth (resp. analytic) SO0(p,q)-actions on Sp+q-1 (p,q≥3) whose restricted SO(p)×SO(q)-action is the standard action. Consequently, we shall show that for each positive integer m there exist uncountably infinite smooth conjugacy classes of smooth SO0(p,q)-actions on Sp+q-1 (p,q≥3) with m closed and m+1 open orbits whose restricted SO(p)×SO(q)-action is the standard action. On the other hand, we shall show that there exist exactly countably many analytic conjugacy classes of analytic SO0(p,q)-actions on Sp+q-1 (p,q≥3) with one closed and two open orbits whose restricted SO(p)×SO(q)-action is the standard action.

Center foliation rigidity for partially hyperbolic toral diffeomorphisms

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over \(\mathbb C\) and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible L has smooth center foliation and is bi-Hölder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Hölder conjugacy must be smooth. For a totally irreducible L with two-dimensional center, we establish a number of equivalent conditions on the perturbation that ensure smooth conjugacy to L.

Topologically conjugate classifications of the translation actions on compact connected Lie groups SU(2) × Tn

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups . We define the rotation vectors of the left actions induced by the elements in the maximal tori of , and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered.

Smooth rigidity for very non-algebraic expanding maps

We show that the space of expanding maps contains an open and dense subset where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.

Learning Dynamics by Reservoir Computing (In Memory of Prof. Pavol Brunovský)

AbstractWe study reservoir computing, a machine learning method, from the viewpoint of learning dynamics. We present numerical results of learning the dynamics of the logistic map, one of the typical examples of chaotic dynamical systems, using a 30-node reservoir and a three-node reservoir. When the learning is successful, an attractor that is smoothly conjugate to the logistic map to be learned is observed in the phase space of the reservoir. Inspired by this numerical result, we introduce a degenerate reservoir system and use it to mathematically confirm this observation. We also show that reservoir computing can learn information about dynamics not included in the training data, which we believe is a remarkable feature of reservoir computing compared to other machine learning methods. We discuss this feature in connection with the above observation that there is a smooth conjugacy between the attractor in the reservoir and the dynamics to be learned.

Global rigidity of some abelian-by-cyclic group actions on 𝕋2

For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild assumptions. Along the way, we also prove a Tits alternative type theorem for some groups of diffeomorphisms of $\T^2$.

Smooth conjugacy of difference equations derived from elliptic curves

We describe the dynamics associated with a construction of tangent lines to elliptic curves which depend on two parameters. With the exception of a half-curve in the parameter plane, the natural compactifications of these maps are shown to be smooth (i.e. ) conjugate to each other and to the classic chaotic map (Chebyshev polynomial) . This provides a new non-trivial example of a class of maps which are smooth conjugate to each other.

LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS

AbstractFollowing Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

Global graph of metric entropy on expanding Blaschke products

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3) × S1 and Spinℂ(3). We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p, −1) × S1 → L(p, −1) × S1, the induced isomorphism (π ◦ f ◦ i)* maps each element in the fundamental group of L(p, −1) to itself or its inverse, where i : L(p, −1) → L(p, −1) × S1 is the natural inclusion and π : L(p, −1) × S1 → L(p, −1) is the projection.

Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in {mathbb{T} ^{d} times SU(2)}

We continue our study of the local theory for quasiperiodic cocycles in {mathbb{T} ^{d} times G} , where {G=SU(2)} , over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the space orthogonal to {L^{2}(mathbb{T} ^{d}) hookrightarrow L^{2}(mathbb{T} ^{d}times G)} . Subsequently, we describe the equivalence classes of cocycles under smooth conjugacy, as a function of the parameters defining their K.A.M. normal form. Finally, we derive a complete classification of the dynamics of one-frequency (d = 1) cocycles over a recurrent Diophantine rotation.

Takens theorem for random dynamical systems

In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov exponents.

Anosov Diffeomorphisms and $${\gamma}$$ γ -Tilings

We consider a toral Anosov automorphism \({G_\gamma:{\mathbb{T}}_\gamma\to{\mathbb{T}}_\gamma}\) given by \({G_\gamma(x,y)=(ax+y,x)}\) in the \({ }\) base, where \({a\in\mathbb{N} \backslash\{1\}}\), \({\gamma=1/(a+1/(a+1/\ldots))}\), \({v=(\gamma,1)}\) and \({w=(-1,\gamma)}\) in the canonical base of \({{\mathbb{R}}^2}\) and \({{\mathbb{T}}_\gamma={\mathbb{R}}^2/(v{\mathbb{Z}} \times w{\mathbb{Z}})}\). We introduce the notion of \({\gamma}\)-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of \({G_\gamma}\); (ii) affine classes of \({\gamma}\)-tilings; and (iii) \({\gamma}\)-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.

Center foliation: absolute continuity, disintegration and rigidity

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Perturbations of Smooth Actions with Nontrivial Cohomology

We prove a general perturbation result for smooth Lie group actions with nontrivial finite‐dimensional cohomology. It describes sufficient conditions on cohomology over an action which imply that the action lies in a finite‐dimensional family of actions such that any small perturbation of the family intersects the smooth conjugacy class of the given action. We cast the classical KAM result on perturbations of Diophantine vector fields on tori into this general setup, and we address a few applications and potential applications of this result to homogeneous Lie group actions with finite‐dimensional first cohomology. © 2014 Wiley Periodicals, Inc.

Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number

We prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.

Some Qualitative Properties of Lyapunov Exponents for Almost Periodic Hamiltonian Systems

The Lyapunov characteristic exponents play a crucial role in the description of the behavior of dynamical systems. They measure the average rate of convergence or divergence of orbits starting from nearby initial points. Therefore, they can be used to analyze the stability of limits sets and check sensitive dependence on initial conditions, that is, the presence of chaotic attractors. In this paper we explore some properties of Lyapunov exponents of autonomous almost periodic Hamiltonian systems. We study the behavior of the Lyapunov exponents under smooth conjugacy. We show that, due to almost periodicity, the Lyapunov exponents of the almost periodic Hamiltonian system are invariant under a smooth conjugacy. In addition, we obtain effective estimates of Lyapunov exponents by the mean value of the largest eigenvalue of the Lie Bracket \(\left[ J,H^{\prime \prime }\right] \). Finally, two numerical examples are given to indicate the validity of the obtained estimates.

On smooth conjugacy of hyperbolic diffeomorphism germs

We prove that on Rn, under the given algebraic conditions imposed on eigenvalues of their linear approximation operators, any two hyperbolic diffeomorphism germs with generic nonlinear parts are C1 conjugate to each other if and only if their linear parts are similar.

Geometric rigidity of $\times m$ invariant measures

Let \mu be a probability measure on [0,1] which is invariant and ergodic for T_{a}(x)=ax\bmod1 , and 0<\dim\mu<1 . Let f be a local diffeomorphism on some open set. We show that if E\subseteq\mathbb{R} and (f\mu)|_{E}\sim\mu|_{E} , then f'(x)\in\{\pm a^{r}\,:\, r\in\mathbb{Q}\} at \mu -a.e. point x\in f^{-1}E . In particular, if g is a piecewise-analytic map preserving \mu then there is an open g -invariant set U containing supp \mu such that g|_{U} is piecewise-linear with slopes which are rational powers of a . In a similar vein, for \mu as above, if b is another integer and a,b are not powers of a common integer, and if \nu is a T_{b} -invariant measure, then f\mu\perp\nu for all local diffeomorphisms f of class C^{2} . This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of T_{a},T_{b} is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if U,V are maps of \mathbb{R}/\mathbb{Z} which are C^{2} -conjugate to T_{a},T_{b} then they have no common measures of positive dimension which are ergodic for both.