Carlos Gustavo Moreira

Stable intersections of Cantor sets and homoclinic bifurcations

Abstract We study intersections of dynamically defined Cantor sets and consequences to dynamical systems. The concept of stable intersection of two dynamically defined Cantor sets is introduced. We prove that if the stable and unstable Cantor sets associated to a homoclinic bifurcation have a stable intersection, then there are open sets in the parameter line with positive density at the initial bifurcating value, for which the corresponding diffeomorphisms are not hyperbolic. We present conditions more general than the ones previously known that assure stable intersections. We also present conditions for hyperbolicity to be of positive density at homoclinic bifurcations. This allow us to provide persistent one-parameter families of homoclinic bifurcations that present both hyperbolicity and homoclinic tangencies with positive density at the initial bifurcating value in the parameter line.

Measures of Pseudorandomness for Finite Sequences: Minimal Values

Mauduit and Sarkozy introduced and studied certain numerical parameters associated to finite binary sequences

Hausdorff measures and the Morse-Sard theorem

E_N\in\{-1,1\}^N

On asymptotic estimates for arithmetic cost functions

in order to measure their ‘level of randomness’. Two of these parameters are the normality measure

From discrete- to continuous-time ergodic theorems

\cal{N}(E_N)

Bifurcation of the essential dynamics of Lorenz maps and applications to Lorenz-like flows: Contributions to the study of the expanding case

and the correlation measure

Differentiable structures of central Cantor sets

C_k(E_N)

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

of order k, which focus on different combinatorial aspects of

BIFURCATION OF THE ESSENTIAL DYNAMICS OF LORENZ MAPS ON THE REAL LINE AND THE BIFURCATION SCENARIO FOR LORENZ LIKE FLOWS: THE CONTRACTING CASE

E_N

Maximal invariant sets for restrictions of tent and unimodal maps

. In their work, amongst others, Mauduit and Sarkozy investigated the minimal possible value of these parameters.In this paper, we continue the work in this direction and prove a lower bound for the correlation measure