Cantor Sets with Complicated Geometry and Modeled by General Symbolic Dynamics

On the Dimension of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, and the Eckmann-ruelle Conjecture

In this paper we unify and extend many of the known results on the dimensions of deterministic and random Cantor-like sets in R n using their symbolic representation. We also construct several new examples of such constructions that illustrate some new phenomena. These sets are deened by geometric constructions with arbitrary placement of subsets. We consider Markov constructions, general symbo...

A Non-additive Thermodynamic Formalism and Applications to Dimension Theory of Hyperbolic Dynamical Systems

A non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they are coded by an arbitrary symbolic dynamics, and the geometry of the construction may depend on all the s...

Dimension of Cantor Sets with Complicated Geometry

Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations

In [12], we gave an intensive study on the level sets of the integrable cubic nonlinear Schrödinger (NLS) equation. Based upon that study, the existence of a symmetric pair of homoclinic orbits in certain perturbed NLS systems was established in [11]. [Stated in Theorem 1.3 below.] In this paper, the existence of Smale horseshoes and symbolic dynamics is established in the neighborhood of the s...

Definability, automorphisms, and dynamic properties of computably enumerable sets

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of FelixKlein’sErlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and theMartin Invariance Conjecture on their Tur...