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Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type  [PDF]
Jacobo Pejsachowicz,Nils Waterstraat
Mathematics , 2013,
Abstract: Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.
Bifurcation of Fredholm Maps I; The Index Bundle and Bifurcation  [PDF]
Jacobo Pejsachowicz
Mathematics , 2010,
Abstract: We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $\beta(f)$ is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized $J$-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.
On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems  [PDF]
Wiktor Radzki
Mathematics , 2008, DOI: 10.1088/0951-7715/23/8/002
Abstract: This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].
Hausdorff dimension of the set of extreme points of a self-similar set  [PDF]
Andrew Tetenov,Ivan Davydkin
Mathematics , 2002,
Abstract: If the system S of contracting similitudes on $ R^2$ satisfies open convex set condition, then the set F of extreme points of the convex hull $\tilde{K}$ of it's invariant self-similar set K has Hausdorff dimension 0 . If, additionally, all the rotation angles of the similitudes are commensurable with $\pi$, then the set F is finite and the convex hull $\tilde{K}$ is a convex polygon.
Bifurcation Results for a Class of Perturbed Fredholm Maps  [cached]
Benevieri Pierluigi,Calamai Alessandro
Fixed Point Theory and Applications , 2008,
Abstract: We prove a global bifurcation result for an equation of the type , where is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset of , are and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions with , whose closure contains a trivial solution . The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called -Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.
Bifurcation Results for a Class of Perturbed Fredholm Maps  [cached]
Pierluigi Benevieri,Alessandro Calamai
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/752657
Abstract: We prove a global bifurcation result for an equation of the type Lx+ (h(x)+k(x))=0, where L:E ¢ € ‰ ¢ € ‰ ¢ ’ ¢ € ‰ ¢ € ‰F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset of E, h,k: —[0,+ ¢ ) ¢ € ‰ ¢ € ‰ ¢ ’ ¢ € ‰ ¢ € ‰F are C1 and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions (x, ) with ¢ ‰ 0, whose closure contains a trivial solution (x ˉ,0). The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called ±-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.
Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension  [PDF]
Eugeniusz Dymek
Mathematics , 2013,
Abstract: For each irrational $\alpha\in[0,1)$ we construct a continuous function $f\: [0,1)\to \R$ such that the corresponding cylindrical transformation $[0,1)\times\R \ni (x,t) \mapsto (x+\alpha, t+ f(x)) \in [0,1)\times\R$ is transitive and the Hausdorff dimension of the set of points whose orbits are discrete is 2. Such cylindrical transformations are shown to display a certain chaotic behaviour of Devaney-like type.
The Hausdorff dimension of the set of dissipative points for a Cantor-like model set for singly cusped parabolic dynamics  [PDF]
J. Schmeling,B. O. Stratmann
Mathematics , 2008,
Abstract: In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.
Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space  [PDF]
Yitwah Cheung
Mathematics , 2006,
Abstract: In this paper we compute the Hausdorff dimension of the set D_n of points on divergent trajectories of the homogeneous flow induced by a certain one-parameter subgroup of G=SL(2,R) acting by left multiplication on the product space G^n/Gamma^n, where Gamma=SL(2,Z). We prove that the Hausdorff dimension of D_n equals 3n-(1/2) for any n greater than one.
Multifractal analysis of dimension spectrum and the set of irregular points in non-uniformly hyperbolic systems  [PDF]
Ma Guan-Zhong,Yao Xiao
Mathematics , 2014,
Abstract: We study the multifractal analysis of dimension spectrum for almost additive potential in a class of one dimensional non-uniformly hyperbolic dynamic systems and prove that the irregular set has full Hausdroff dimension.
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