ISSN:

1930-5311

eISSN:

1930-532X

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## Journal of Modern Dynamics

2015 , Volume 9

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2015, 9: 1-23
doi: 10.3934/jmd.2015.9.1

*+*[Abstract](1524)*+*[PDF](254.6KB)**Abstract:**

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of $SL(2,\mathbb R)$ on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.

2015, 9: 25-49
doi: 10.3934/jmd.2015.9.25

*+*[Abstract](1612)*+*[PDF](259.3KB)**Abstract:**

We effectively conclude the local rigidity program for generic restrictions of partially hyperbolic Weyl chamber flows. Our methods replace and extend previous ones by circumventing computations made in Schur multipliers. Instead, we construct a natural topology on $H_2(G,\mathbb{Z})$, and rely on classical Lie structure theory for central extensions.

2015, 9: 51-66
doi: 10.3934/jmd.2015.9.51

*+*[Abstract](1485)*+*[PDF](839.3KB)**Abstract:**

We find optimal upper bounds for spectral invariants of a Hamiltonian whose support is contained in a union of mutually disjoint displaceable balls. This gives a partial answer to a question posed by Leonid Polterovich in connection with his recent work on Poisson bracket invariants of coverings.

2015, 9: 67-80
doi: 10.3934/jmd.2015.9.67

*+*[Abstract](1771)*+*[PDF](242.5KB)**Abstract:**

This work is partially supported by the ERC starting grant GA 257110 “RaWG”. We show that every Grigorchuk group $G_\omega$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; moreover it can have finitely generated infinite torsion subgroups, answering a question of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_\omega$ can be embedded in a finitely generated simple group that has trivial Poisson boundary for every simple random walk.

This work is partially supported by the ERC starting grant GA 257110 “RaWG”.

2015, 9: 81-121
doi: 10.3934/jmd.2015.9.81

*+*[Abstract](1610)*+*[PDF](869.6KB)**Abstract:**

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.

2015, 9: 123-140
doi: 10.3934/jmd.2015.9.123

*+*[Abstract](1344)*+*[PDF](205.6KB)**Abstract:**

We calculate the action of the group of affine diffeomorphisms on the relative cohomology of pair $(M,\Sigma)$, where $M$ is a square-tiled surface that is a normal abelian cover of the flat pillowcase. As an application, we answer a question raised by Smillie and Weiss.

2015, 9: 141-146
doi: 10.3934/jmd.2015.9.141

*+*[Abstract](1631)*+*[PDF](131.7KB)**Abstract:**

We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field on it, there exists a closed magnetic geodesic for almost all kinetic energy levels.

2015, 9: 147-167
doi: 10.3934/jmd.2015.9.147

*+*[Abstract](2242)*+*[PDF](310.0KB)**Abstract:**

We consider reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces with large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has conjugate points along every closed geodesic. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle. The analogous result applies to monotone twist maps.

2015, 9: 169-190
doi: 10.3934/jmd.2015.9.169

*+*[Abstract](1710)*+*[PDF](437.0KB)**Abstract:**

We investigate the notion of complex rotation number which was introduced by V. I. Arnold in 1978. Let $f：\mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ be a (real) analytic orientation preserving circle diffeomorphism and let $\omega\in \mathbb{C}/\mathbb{Z}$ be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus {$z\in \mathbb{C}/\mathbb{Z} | 0 < Im(z)< Im(\omega)$} via the map $f+\omega$. This complex torus is isomorphic to $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ for some appropriate $\tau\in \mathbb{C}/\mathbb{Z}$.

According to Moldavskis [6], if the ordinary rotation number rot$(f+\omega_0)$ is Diophantine and if $\omega$ tends to $\omega_0$ non tangentially to the real axis, then $\tau$ tends to rot$(f+\omega_0)$. We show that the Diophantine and non tangential assumptions are unnecessary: If rot$(f+\omega_0)$ is irrational, then $\tau$ tends to rot$(f+\omega_0)$ as $\omega$ tends to $\omega_0$.

This, together with results of N. Goncharuk [4], motivates us to introduce a new fractal set (``bubbles'') given by the limit values of $\tau$ as $\omega$ tends to the real axis. For the rational values of rot $(f+\omega_0)$, these limits do not necessarily coincide with rot $(f+\omega_0)$ and form a countable number of analytic loops in the upper half-plane.

2015, 9: 191-201
doi: 10.3934/jmd.2015.9.191

*+*[Abstract](1473)*+*[PDF](171.4KB)**Abstract:**

We show the local rigidity of the standard action of the Borel subgroup of $SO_+(n,1)$ on a cocompact quotient of $SO_+(n,1)$ for $n \geq 3$.

2015, 9: 203-218
doi: 10.3934/jmd.2015.9.203

*+*[Abstract](1480)*+*[PDF](476.2KB)**Abstract:**

We study the intersection of a positively sectional-hyperbolic set and a negatively sectional-hyperbolic set of a flow on a compact manifold. Indeed, we show that such an intersection is not a hyperbolic set in general. Next, we show that such an intersection is a hyperbolic set if the sets involved in the intersection are both transitive. In general, we prove that such an intersection is the disjoint union of a nonsingular hyperbolic set, a finite set of singularities, and a set of regular orbits joining these dynamical objects. Finally, we exhibit a three-dimensional star flow with a positively (but not negatively) sectional-hyperbolic homoclinic class and a negatively (but not positively) sectional-hyperbolic homoclinic class whose intersection is a periodic orbit. This provides a counterexample to a conjecture of Shi, Zhu, Gan and Wen ([25], [26]).

2015, 9: 219-235
doi: 10.3934/jmd.2015.9.219

*+*[Abstract](1486)*+*[PDF](302.9KB)**Abstract:**

Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1 (M) \to \mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a given homology class and can be seen as a symplectic analogue of the Riemannian length spectrum. When M has genus zero we also construct Hofer- and $C^0$-continuous quasimorphisms $Ham(M) \to H_1(M)$ that compute trajectories of periodic non-displaceable disks.

2015, 9: 237-255
doi: 10.3934/jmd.2015.9.237

*+*[Abstract](1993)*+*[PDF](259.6KB)**Abstract:**

We give a new proof of a multiplicative ergodic theorem for quasi-compact operators on Banach spaces with a separable dual. Our proof works by constructing the finite-codimensional `slow' subspaces (those where the growth rate is dominated by some $\lambda_i$), in contrast with earlier infinite-dimensional multiplicative ergodic theorems which work by constructing the finite-dimensional fast subspaces. As an important consequence for applications, we are able to get rid of the injectivity requirements that appear in earlier works.

2015, 9: 257-284
doi: 10.3934/jmd.2015.9.257

*+*[Abstract](1467)*+*[PDF](272.9KB)**Abstract:**

Given a word $w$ on $n$ letters, we study groups which satisfy ``iterated identity'' $w$, meaning that for all $x_1, \dots, x_n$ there exists $N$ such that the $N-th$ iteration of $w$ of Engel type, applied to $x_1, \dots, x_n$, is equal to the identity. We define bounded groups and groups which are multiscale with respect to identities. This notion of being multiscale can be viewed as a self-similarity conditions for the set of identities, satisfied by a group. In contrast with torsion groups and Engel groups, groups which are multiscale with respect to identities appear among finitely generated elementary amenable groups. We prove that any polycyclic, as well as any metabelian group is bounded, and we compute the iterational depth for various wreath products. We study the set of iterated identities satisfied by a given group, which is not necessarily a subgroup of a free group and not necessarily invariant under conjugation, in contrast with usual identities. Finally, we discuss another notion of iterated identities of groups, which we call solvability type iterated identities, and its relation to elementary classes of varieties of groups.

2015, 9: 289-304
doi: 10.3934/jmd.2015.9.289

*+*[Abstract](1524)*+*[PDF](205.0KB)**Abstract:**

We prove that there exists an interval exchange transformation and a point so that the orbit of the point equidistributes according to a non-ergodic measure. That is, it is possible for a non-ergodic measure to arise from the Krylov-Bogolyubov construction of invariant measures for an interval exchange transformation.

2015, 9: 305-353
doi: 10.3934/jmd.2015.9.305

*+*[Abstract](1858)*+*[PDF](464.6KB)**Abstract:**

We prove quantitative equidistribution results for actions of Abelian subgroups of the $(2g+1)$-dimensional Heisenberg group acting on compact $(2g+1)$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the classical results of Hardy and Littlewood [25,26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.

2015, 9: 357-363
doi: 10.3934/jmd.2015.9.357

*+*[Abstract](1648)*+*[PDF](162.0KB)**Abstract:**

In this paper, we prove: (1) for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense; (2) for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is a $C^\infty$-closing lemma for three-dimensional Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

2015, 9: 365-405
doi: 10.3934/jmd.2015.9.365

*+*[Abstract](1519)*+*[PDF](934.2KB)**Abstract:**

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the usual regional proximality relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have less intricate dynamics, and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions and a computation of the automorphism group of the minimal Robinson tiling.

2018 Impact Factor: 0.295

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