Divergent Orbits Research Articles

Closures of locally divergent orbits of maximal tori and values of homogeneous forms

AbstractLet ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

Bounded and divergent trajectories and expanding curves on homogeneous spaces

Suppose g t g_t is a 1 1 -parameter A d \mathrm {Ad} -diagonalizable subgroup of a Lie group G G and Γ > G \Gamma > G is a lattice. We study the dimension of bounded and divergent orbits of g t g_t emanating from a class of curves lying on leaves of the unstable foliation of g t g_t on the homogeneous space G / Γ G/\Gamma . We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of S L ( 2 , R ) \mathrm {SL}(2,\mathbb {R}) inside G G , which are not contained in a proper parabolic subgroup of G G . We describe applications of our results to problems in Diophantine approximation by number fields and intrinsic Diophantine approximation on spheres. Our methods also yield the following result for lines in the space of square systems of linear forms: suppose φ ( s ) = s Y + Z \varphi (s) = sY + Z where Y ∈ G L ( n , R ) Y\in \mathrm {GL}(n,\mathbb {R}) and Z ∈ M n , n ( R ) Z\in M_{n,n}(\mathbb {R}) . Then, the dimension of the set of points s s such that φ ( s ) \varphi (s) is singular is at most 1 / 2 1/2 while badly approximable points have Hausdorff dimension equal to 1 1 .

Unravelling the distinctive craniomandibular morphology of the Plio‐PleistoceneEumysopsin the evolutionary setting of South American octodontoid rodents (Hystricomorpha)

AbstractEchimyidae is a species‐rich clade of Neotropical rodents, which diversified in association with forested biomes. Since the late Miocene, a few lineages from southern South America have been adapted to open environments.Eumysopsis one of these southern echimyids, and its peculiar craniomandibular morphology has been assumed to be a result of adaptation to open environments. We performed a geometric morphometric analysis of craniomandibular shape variation to explore whether, as suspected,Eumysopsis divergent from other echimyids and octodontoids. In addition, we explored whether deterministic factors driven by different ecological dimensions can explain the diversification of shape among octodontoids. We found that craniomandibular shape variation in octodontoids was related to ecological variables. Comparing competing evolutionary models suggested that the input of selective factors play a key role in octodontoid craniomandibular shape diversification; habitat and habits were found to be the most influential factors. In the analysed morphospaces,Eumysopswas located distant from other echimyids due to its distinctive traits, especially wide and posteriorly displaced orbits, and related low craniomandibular joint. Divergent orbits and resulting wider panoramic vision support the interpretation ofEumysopsas an open‐habitat specialist echimyid. But what is more relevant, is thatEumysopsoccupied a sector of the octodontoid cranial morphospace not filled by living representatives; this highlights the contribution of fossils in providing key information on the specialization boundaries explored by a clade throughout its history.

Equidistribution of divergent orbits of the diagonal group in the space of lattices

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: the discriminant—an integer—and the type—an integer vector. We then study the question of the limit distributional behavior of these orbits as the discriminant goes to infinity. Using entropy methods we prove that, for divergent orbits of a specific type, virtually any sequence of orbits equidistributes as the discriminant goes to infinity. Using measure rigidity for higher-rank diagonal actions, we complement this result and show that, in dimension three or higher, only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.

Equidistribution of divergent orbits and continued fraction expansion of rationals

We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis we obtain new results regarding the asymptotic normality of the continued fraction expansions of most rationals with a high denominator as well as an estimate on the length of their continued fraction expansions. By similar methods we also establish a complementary result to Zaremba's conjecture. Namely, we show that given a bound M, for any large q, the number of rationals $p/q\in [0,1]$ for which the coefficients of the continued fraction expansion of p/q are bounded by M is $o(q^{1-\epsilon})$ for some $\epsilon>0$ which depends on M.

Singular systems of linear forms and non-escape of mass in the space of lattices

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.

Averaging structure in the [formula omitted] problem

The 3x+1 problem has resisted analysis from multiple perspectives for many decades. This paper studies the more general qx+r problem, where q and r are odd, and finds new, averaging structures for the iterates. This structure supports the conjecture that all orbits enter a cycle if q=1,3 but divergent orbits exist if q≥5.

Rethinking Primate Origins Again

In 1974, Cartmill introduced the theory that the earliest primate adaptations were related to their being visually oriented predators active on slender branches. Given more recent data on primate-like marsupials, nocturnal prosimians, and early fossil primates, and the context in which these primates first appeared, this theory has been modified. We hypothesize that our earliest primate relatives were likely exploiting the products of co-evolving angiosperms, along with insects attracted to fruits and flowers, in the slender supports of the terminal branch milieu. This has been referred to as the primate/angiosperm co-evolution theory. Cartmill subsequently posited that: "If the first euprimates had grasping feet and blunt teeth adapted for eating fruit, but retained small divergent orbits…" then the angiosperm coevolution theory would have support. The recent discovery of Carpolestes simpsoni provides this support. In addition, new field data on small primate diets, and a new theory concerning the visual adaptations of primates, have provided further evidence supporting the angiosperm coevolution theory.

Locally Divergent Orbits on Hilbert Modular Spaces

We describe the closures of locally divergent orbits under the action of tori on Hilbert modular spaces of rank r≥2. In particular, we prove that if D is a maximal -split torus acting on a real Hilbert modular space, then every locally divergent nonclosed orbit is dense for r>2 and its closure is a finite union of tori orbits for r=2. Our results confirm an orbit rigidity conjecture of G. A. Margulis in all cases except for (i) r=2 and (ii) r>2 and the Hilbert modular space corresponds to a CM-field; in the cases (i) and (ii), our results contradict the conjecture. As an application, we describe the set of values at integral points of collections of nonproportional, split, binary, quadratic forms over number fields.

Fiberwise divergent orbits of projective flows with exactly two minimal sets

Let ’t be a nonsingular flow on a 3-dimensional manifold M. Denote by P : PX ! M the projectivized bundle of the quotient bundle of TM by the line bundle tangent to’t. The derivative of’t induces a flow t on PX, called the projective flow of ’t. In this paper, we consider the dynamical properties of t restricted to 1 P (M) for a minimal set M of’t, under the condition that the restriction of t to 1 P (M) has exactly two minimal sets N1 and N2. If’t has no dominated splitting over M, we find two types of orbits of t in the domain between N1 and N2: one is “bounded below” and the other is “bounded above”. As an application we prove that, if’t is further assumed to be almost periodic on the minimal set, there is a dense orbit in that domain.

Divergent Orbits on S-adic Homogeneous Spaces

Let G be a semisimple algebraic group defined over a number field K and let S be a finite set of non-equivalent valuations of K containing the archimedean ones. Set G = ∏ v∈S G(Kv) and Γ = G(O) where O is the ring of S-integers of K. Fix v ∈ S and a Kv-split algebraic torus Tv of G(Kv). In this paper, in complement to results from [To], we prove results about the divergent orbits for the action of Tv on G/Γ by left translation.

The sufficiency of arithmetic progressions for the 3𝑥+1 Conjecture

Define T : Z + → Z + T:\mathbb {Z} ^{+}\rightarrow \mathbb {Z} ^{+} by T ( x ) = ( 3 x + 1 ) / 2 T\left ( x\right ) =\left ( 3x+1\right ) /2 if x x is odd and T ( x ) = x / 2 T\left ( x\right ) =x/2 if x x is even. The 3 x + 1 3x+1 Conjecture states that the T T -orbit of every positive integer contains 1 1 . A set of positive integers is said to be sufficient if the T T -orbit of every positive integer intersects the T T -orbit of an element of that set. Thus to prove the 3 x + 1 3x+1 Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets 1 + 2 n N 1+2^{n} \mathbb {N} are sufficient for n ≤ 4 n\leq 4 and asked if 1 + 2 n N 1+2^{n}\mathbb {N} is also sufficient for larger values of n n . We answer this question in the affirmative by proving the stronger result that A + B N A+B\mathbb {N} is sufficient for any nonnegative integers A A and B B with B ≠ 0 , B\neq 0, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

Spurs and feathering in spiral galaxies

We present smoothed particle hydrodynamic (SPH) simulations of the response of gas discs to a spiral potential. These simulations show that the commonly observed spurs and feathering in spiral galaxies can be understood as being due to structures present in the spiral arms that are sheared by the divergent orbits in a spiral potential. Thus, dense molecular cloud-like structures generate the perpendicular spurs as they leave the spiral arms. Subsequent feathering occurs as spurs are further sheared into weaker parallel structures as they approach the next spiral passage. Self-gravity of the gas is not included in these simulations, stressing that these features are purely due to the hydrodynamics in spiral shocks. Instead, a necessary condition for this mechanism to work is that the gas need be relatively cold (1000 K or less) in order that the shock is sufficient to generate structure in the spiral arms, and such structure is not subsequently smoothed by the gas pressure.

LOCAL ACTIVITY OF THE VAN DER POL CNN

This paper studies a class of coupled Van der Pol (CVDP) cellular neural networks (CNNs) that can be realized via a coupled fourth-order circuit with two synaptic currents. The local activity theory, developed by Chua in 1997, is applied to study the CVDP CNN, thereby revealing that the bifurcation diagram of the CVDP CNN has a local activity domain with an edge of chaos, as well as a one-dimensional locally passive domain. Although no chaotic phenomena have been identified in simulations, many complex dynamical behaviors have been observed, such as the co-existence of one-periodic, divergent, and convergent orbits, at the edge of chaos.

Closed orbits for actions of maximal tori on homogeneous spaces

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be any torus containing a maximal $\mathbb{R}$-split torus. We prove that the closed orbits for the action of $T$ on $G/\Gamma$ admit a simple algebraic description. In particular, we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is a product of a compact torus and a torus defined over $\mathbb{Q}$, and it is divergent if and only if the maximal $\mathbb{R}$-split subtorus of $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following: · there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit; · if $\rank_{\mathbb{Q}}G<\rank_{\mathbb{R}} G$, there are no divergent orbits for $T$.

New Riddling Bifurcation in Asymmetric Dynamical Systems

We investigate the bifurcation mechanism for the loss of transverse stability of the chaotic attractor in an invariant subspace in an asymmetric dynamical system. It is found that a direct transition to global riddling occurs through a transcritical contact bifurcation between a periodic saddle embedded in the chaotic attractor on the invariant subspace and a repeller on its basin boundary. This new bifurcation mechanism differs from that in symmetric dynamical systems. After such a riddling bifurcation, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and typical power-law scaling is found.

Mechanism for the riddling transition in coupled chaotic systems

We investigate the loss of chaos synchronization in coupled chaotic systems without symmetry from the point of view of bifurcations of unstable periodic orbits embedded in the synchronous chaotic attractor (SCA). A mechanism for direct transition to global riddling through a transcritical contact bifurcation between a periodic saddle embedded in the SCA and a repeller on the boundary of its basin of attraction is thus found. Note that this bifurcation mechanism is different from that in coupled chaotic systems with symmetry. After such a riddling transition, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and thus typical power-law scaling is found.

On the interaction of tropical‐cyclone‐scale vortices. IV: Baroclinic vortices

AbstractThe binary interaction of tropical cyclones is investigated using a three‐dimensional primitive‐equation model. The extended anticyclonic circulations in the upper troposphere merge at very large vortex‐separation distances. For the cyclonic component in the lower troposphere, we find three fundamental modes of interaction separated by two critical separation distances: a mutual approach separation (MAS) and a mutual merger separation (MMS). We suggest that failure to identify these modes may have caused some confusion in interpreting previous baroclinic interaction studies.The MAS delineates vortices that approach each other from those that move on divergent orbits. The approach phase consists of steady radial movement and gradual acceleration, with deformation of the outer vorticity structure of each vortex but little change to their cores. In contrast to barotropic studies, the MAS is much larger than the radius at which the potential‐vorticity gradient of each vortex changes sign. Vortex tilting associated with the vertical shear of the azimuthal winds from the opposing vortex and secondary circulations associated with diabatic heating increases the mutual vortex attraction. The presence of an earth‐vorticity gradient reduces this attraction slightly, but also introduces considerable sensitivity to vortex orientation. When all physical processes are included, we find an MAS of around 1000 km with a scatter of several hundred km, which agrees well with observational studies.Approach occurs with little change in the vortex cores until they reach the MMS. Merger then occurs very rapidly, usually within several hours, and follows that described in parts II and III for barotropic vortices. The MMS is approximately three times the equivalent vortex‐patch radius for the cyclones; it is slightly reduced by diabatic heating, but it is largely independent of the earth‐vorticity gradient. The vortices experience a slight weakening during the approach and initial merger stages. However, with diabatic heating, rapid intensification follows merger; such intensification may have implications for rapid development of tropical cyclones.