Dimension and product structure of hyperbolic measures.
Dimension groups for interval maps.
Dimension of a measure
We propose a framework to define dimensions of Borel measures in a metric space by formulating a set of natural properties for a measure-dimension mapping, namely monotonicity, bi-Lipschitz invariance, (σ-)stability, etc. We study the behaviour of most popular definitions of measure dimensions in regard to our list, with special attention to the standard correlation dimensions and their modified versions.
Dimension of countable intersections of some sets arising in expansions in non-integer bases
We consider expansions of real numbers in non-integer bases. These expansions are generated by β-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.
Dimension of measures: the probabilistic approach.
Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.
Dimension of weakly expanding points for quadratic maps
For the real quadratic map and a given a point has good expansion properties if any interval containing also contains a neighborhood of with univalent, with bounded distortion and for some . The -weakly expanding set is the set of points which do not have good expansion properties. Let denote the negative fixed point and the first return time of the critical orbit to . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff dimension of...
Dimension product structure of hyperbolic sets.
Dimensions des spirales
Dimensions of Prym varieties.
Dimensions of subsets of Cantor-type sets.
Dimensions of the boundaries of self-similar sets.
Dimensions of the Julia sets of rational maps with the backward contraction property
Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs and areequivalentif the Newton polygons of the corresponding partition functions...
Diophantine classes, dimension and Denjoy maps
Diophantine conditions in global well-posedness for coupled KdV-type systems.
Directional transition matrix
We present a generalization of topological transition matrices introduced in [6].
Dirichlet forms on quotients of shift spaces
We define thin equivalence relations ∼ on shift spaces and derive Dirichlet forms on the quotient space in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.
Discontinuous Hamiltonian flows for nonlinear control system.
Discrete anisotropic curvature flow of graphs
Discrete anisotropic curvature flow of graphs
The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfaces in natural norms. Furthermore we prove a global existence result for the continuous problem and present some examples of computed surfaces.