Authors:Xavier Carvajal; Amin Esfahani; Mahendra Panthee Pages: 505 - 550 Abstract: Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0034-z Issue No:Vol. 48, No. 4 (2017)

Authors:Ryuichi Fukuoka; Djeison Benetti Pages: 551 - 598 Abstract: Let G be a group, (M, d) be a metric space, \(X\subset M\) be a compact subset and \(\varphi :G\times M\rightarrow M\) be a left action of G on M by homeomorphisms. Denote \(gp=\varphi (g,p)\) . The isotropy subgroup of G with respect to X is defined by \(H_X=\{g\in G; gX=X\}\) . In this work we define the induced Hausdorff metric on \(G/H_X\) by \(d_X(g_1H_X,g_2H_X):=d_H(g_1X,g_2X)\) , where \(d_H\) is the Hausdorff distance on M. Let \(\hat{d}_X\) be the intrinsic metric induced by \(d_X\) . In this work, we study the geometry of \((G/H_X,d_X)\) and \((G/H_X,\hat{d}_X)\) and their relationship with (M, d). In particular, we prove that if G is a Lie group, M is a differentiable manifold endowed with a metric which is locally Lipschitz equivalent to a Finsler metric, \(X\subset M\) is a compact subset and \(\varphi :G\times M\rightarrow M\) is a smooth left action by isometries, then \((G/H_X,\hat{d}_X)\) is a \(C^0\) -Finsler manifold. We also calculate the Finsler metric explicitly in some examples. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0032-1 Issue No:Vol. 48, No. 4 (2017)

Authors:Noriaki Kawaguchi Pages: 599 - 622 Abstract: We extend the study on shadowable points recently introduced by Morales in relation to chaotic or non-chaotic properties. Firstly, some sufficient conditions for a quantitative shadowable point to be approximated by an entropy point are given. As a corollary, we get different three chaotic conditions from which a shadowable point becomes an entropy point. Secondly, we provide a dichotomy on the interior of the set of shadowable chain recurrent points by two canonical chaotic and non-chaotic dynamics, the full shift and odometers. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0033-0 Issue No:Vol. 48, No. 4 (2017)

Authors:H. Sano; Y. Kabata; J. L. Deolindo Silva; T. Ohmoto Pages: 623 - 639 Abstract: We present a local classification of smooth surfaces in \({\mathbb {P}}^3\) in terms of the singularity types (of codimension \(\le \) 4) of their central projections to a plane. Based on our classification result, we also give exact normal forms to surface germs at transition moments on bifurcations with respect to parabolic curves and flecnodal curves. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0036-x Issue No:Vol. 48, No. 4 (2017)

Authors:Angel Cano; Luis Loeza; Alejandro Ucan-Puc Pages: 641 - 647 Abstract: In this article we show that Bers’ simultaneous uniformization and Köebe’s retrosection theorem do not hold for discrete groups of projective transformations acting on complex projective space. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0035-y Issue No:Vol. 48, No. 4 (2017)

Authors:Dante Carrasco-Olivera; Bernardo San Martín Pages: 649 - 662 Abstract: It is well known that the geometric Lorenz attractor is \(\mathcal {K}^*\) -expansive. In this paper we prove that the Rovella attractor is also \(\mathcal {K}^*\) -expansive in an almost 2-persistent way. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0038-8 Issue No:Vol. 48, No. 4 (2017)

Authors:David A. C. Mollinedo; Christian Olivera Pages: 663 - 677 Abstract: The Cauchy problem for a multidimensional linear transport equation with unbounded drift is investigated. Provided the drift is Holder continuous , existence, uniqueness and strong stability of solutions are obtained. The proofs are based on a careful analysis of the associated stochastic flow of characteristics and techniques of stochastic analysis. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0039-7 Issue No:Vol. 48, No. 4 (2017)

Authors:Masaru Hasegawa; Farid Tari Pages: 679 - 696 Abstract: The simplest way to have birth of surfaces is through transitions in the fibres of a function f with a Morse singularity of index 0 or 3. It is natural to seek to understand the geometry of newly born surfaces. We consider here the question of finding how many umbilics are on a newly born surface. We show that newly born surfaces in the Euclidean 3-space have exactly 4 umbilic points all of type lemon, provided that the Hessian of f at the singular point has pairwise distinct eigenvalues. This is true in both cases when f is an analytic or a smooth germ. When only two of such eigenvalues are equal, the number of umbilic points is either 2, 4, 6 or 8 when f is an analytic or a generic smooth germ. The same results holds for newly born surfaces in the Minkowski 3-space. In that case when the two eigenvalues associated to the two spacelike eigenvectors are distinct we get exactly 4 umbilic points all of type lemon. If they are equal, the number of umbilic points is either 2, 4, 6 or 8. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0037-9 Issue No:Vol. 48, No. 4 (2017)

Authors:Cícero Aquino; Halyson Baltazar Pages: 697 - 715 Abstract: The purpose of this article is to study the uniqueness of complete hypersurfaces satisfying some pinching curvature condition. Here, we use the generalized maximum principle of Omori–Yau to obtain uniqueness results for complete spacelike hypersurfaces immersed in a Lorentzian product space. In addition, we obtain the analogue results for complete hypersurfaces immersed in a Riemannian product space. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0041-0 Issue No:Vol. 48, No. 4 (2017)

Authors:Ítalo Melo Pages: 717 - 732 Abstract: In this paper we will consider the concept of \(\mathbb {P}\) -weakly hyperbolic iterated function systems on compact metric spaces that generalizes the concept of weakly hyperbolic iterated function systems, as defined by Edalat (Inf Comput 124(2):182–197, 1996) and by Arbieto, Santiago and Junqueira (Bull Braz Math Soc New Ser 2016) for a more general setting where the parameter space is a compact metric space. We prove the existence and uniqueness of the invariant measure of a \(\mathbb {P}\) -weakly hyperbolic IFS. Furthermore, we prove an ergodic theorem for \(\mathbb {P}\) -weakly hyperbolic IFS with compact parameter space. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0042-z Issue No:Vol. 48, No. 4 (2017)

Authors:Federico Quallbrunn Pages: 335 - 345 Abstract: Following Suwa, we study unfoldings of algebraic foliations and their relationship with families of foliations, making focus on those unfoldings related to trivial families. The results obtained in the study of unfoldings are then applied to obtain information on the structure of foliations on projective spaces. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0024-6 Issue No:Vol. 48, No. 3 (2017)

Authors:V. Ramos; J. Siqueira Pages: 347 - 375 Abstract: We prove uniqueness of equilibrium states for a family of partially hyperbolic horseshoes associated to a class of Hölder continuous potentials with small variation and derive statistical properties for this unique equilibrium. We define a projection map associated to the horseshoe and prove a spectral gap for its transfer operator acting on some space of Hölder continuous observables. From this we deduce an exponential decay of correlations and a central limit theorem. We finally extend these results to the horseshoe via Rohlin’s disintegration of the equilibrium along the stable fibers. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0027-y Issue No:Vol. 48, No. 3 (2017)

Authors:Nancy Guelman; Isabelle Liousse Pages: 389 - 397 Abstract: A group \(\Gamma \) is said to be periodic if for any g in \(\Gamma \) there is a positive integer n with \(g^n=id\) . We first prove that a finitely generated periodic group acting on the 2-sphere \({\mathbb S}^2\) by \(C^1\) -diffeomorphisms with a finite orbit, is finite and \(C^1\) -conjugate to a subgroup of \(\mathrm {O}(3,{\mathbb R})\) . This result is obtained by proving the more general statement: a finitely generated periodic group acting on any compact manifold by \(C^1\) -diffeomorphisms with a finite orbit, is finite. We use it for proving that a countable 2-group of spherical diffeomorphisms with bounded orders is finite. This gives a negative partial answer to a question posed by D. Fisher. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus is finite. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0028-x Issue No:Vol. 48, No. 3 (2017)

Authors:A. Bagatini; M. L. Matte; A. Wagner Pages: 413 - 437 Abstract: Considering the unsigned version of the mock theta functions \(\phi (q)\) , \(\psi (q)\) , \(f_0(q)\) , \(F_0(q)\) , \(f_1(q)\) and \(F_1(q),\) they represent partitions subject to some rules. Based on a two-line matrix representation, we can classify them according to the sum of second line and derive some results such as closed formulas and identities for other types of partitions. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0021-9 Issue No:Vol. 48, No. 3 (2017)

Authors:Zhengxin Zhou Pages: 439 - 447 Abstract: In this article, I have completely solved this problem: when one differential system is equivalent to a given differential system, what structure does this system and its reflecting integral have' At the same time, I have established the relationship between the reflecting integrals and the first integrals and integrating factors of the differential equations. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0026-4 Issue No:Vol. 48, No. 3 (2017)

Authors:Erica Boizan Batista; João Carlos Ferreira Costa; Juan J. Nuño-Ballesteros Abstract: We consider the topological classification of finitely determined map germs \([f]:(\mathbb {R}^3,0)\rightarrow (\mathbb {R}^2,0)\) with \(f^{-1}(0)\ne \{0\}\) . The case \(f^{-1}(0) = \{0\}\) was treated in another recent paper by the authors. The main tool used to describe the topological type is the link of [f], which is obtained by taking the intersection of its image with a small sphere \(S^1_\delta \) centered at the origin. The link is a stable map \(\gamma _f:N\rightarrow S^1\) , where N is diffeomorphic to a sphere \(S^2\) minus 2L disks. We define a complete topological invariant called the generalized Reeb graph. Finally, we apply our results to give a topological description of some map germs with Boardman symbol \(\Sigma ^{2,1}\) . PubDate: 2017-11-16 DOI: 10.1007/s00574-017-0058-4

Authors:Marcos Teixeira Alves; Jurandir Ceccon; Higidio Portillo Oquendo Abstract: Let (M, g) be a n-dimensional smooth compact Riemannian manifold without boundary with \(n\ge 2\) . We prove that the optimal Riemannian p-entropy inequality $$\begin{aligned} \int _M u ^p\log ( u ^p) \; dv_g\le \dfrac{n}{\tau }\log \left[ {\mathcal {A}}(p,\tau )\left( \int _M \nabla _g u ^p\; dv_g\right) ^{\frac{\tau }{p}}+{\mathcal {B}}(p)\right] \end{aligned}$$ is valid for every \(u\in H^{1,p}(M)\) with \(\Vert u\Vert _p=1\) where \(p>1\) and \(1\le \tau < \min \{2,p\}\) or \(\tau = p \le 2\) . Also, we investigated the relationship between this optimal inequality and the hypercontractivity property for the non-linear evolution equation $$\begin{aligned} u_t=\Delta _p\left( u^{\frac{1}{p-1}}\right) ,\quad x\in M,\ t>0. \end{aligned}$$ When \(\tau =p\le 2\) we find explicit estimates of the time asymptotic behavior of their solutions with initial data in the spaces \(L^q(M)\) , \(1\le q<\infty \) . PubDate: 2017-11-04 DOI: 10.1007/s00574-017-0057-5

Authors:Inderdeep Singh; Sheo Kumar Abstract: Solutions of partial differential equations which are not enough smooth, when approximated by cubic, quadratic and linear polynomials results in poor convergence or no convergence in results. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are more suitable and successful. In view of the above, numerical methods based on Haar wavelet are developed for solving third-order Harry Dym (HD), Benjamin–Bona–Mahony–Burger’s (BBM Burger’s) equation and 2D diffusion equation. Numerical examples have been solved to establish that the present methods give better results than the numerical methods described in past literature. PubDate: 2017-11-02 DOI: 10.1007/s00574-017-0055-7

Authors:Pengcheng Niu; Leyun Wu Abstract: We consider the quasilinear degenerate elliptic equation with rough and singular coefficients of the form $$\begin{aligned} div\left( {A(x,u,\nabla u)} \right) = B(x,u,\nabla u) \end{aligned}$$ in an open set \(\Omega \subset {\mathbb {R}^n},\) where the quadratic form associated with the principle part of this equation allows vanishing and the coefficients in structural conditions on \(A(x,u,\nabla u)\) and \(B(x,u,\nabla u)\) require no smoothness but belong to some Stummel–Kato class. We prove a Fefferman–Phong type inequality related to the Stummel–Kato class and then an embedding inequality. Based on these inequalities, the local boundedness and Harnack’s inequality of the weak solutions are derived. As applications, the continuity and Hölder continuity for the nonnegative weak solutions are given. PubDate: 2017-10-09 DOI: 10.1007/s00574-017-0054-8

Authors:Waldemar Barrera; Adriana Gonzalez-Urquiza; Juan Pablo Navarrete Abstract: In this paper we give a generalization of the Conze–Guivarc’h limit set. With this definition the limit set has very similar properties to those of the limit set in hyperbolic spaces. Moreover, we prove a relation between this new limit set and the Kulkarni limit set. Additionally we show that some closed subsets can be approximated by the Conze–Guivarc’h limit set. This is a result in the theory of classic Kleinian groups. PubDate: 2017-09-02 DOI: 10.1007/s00574-017-0053-9