TY - JOUR
AB - We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case.
AU - Kaloshin, Vadim
AU - Zhang, Ke
ID - 8420
IS - 11
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - Density of convex billiards with rational caustics
VL - 31
ER -
TY - JOUR
AB - In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let ${\mathbb T}^2$ be a 2-dimensional torus and B2 be the unit ball around the origin in ${\mathbb R}^2$ . Fix ρ > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom $H_0(p)+\varepsilon H_1(\theta,p,t),\quad \ \theta\in {\mathbb T}^2,\ p\in B^2,\ t\in {\mathbb T}={\mathbb R}/{\mathbb Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ , namely, a ρ-neighborhood of the orbit contains ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ .
Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].
AU - Kaloshin, Vadim
AU - Zhang, K
ID - 8498
IS - 8
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - Arnold diffusion for smooth convex systems of two and a half degrees of freedom
VL - 28
ER -
TY - JOUR
AB - We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.
AU - Hunt, Brian R
AU - Kaloshin, Vadim
ID - 8527
IS - 5
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - How projections affect the dimension spectrum of fractal measures
VL - 10
ER -