arXiv subjects
Publications and source records attributed to Javier Correa.
We show a flexibility result in the context of generalized entropy. The space of dynamical systems we work with is, homeomorphisms on the sphere whose non-wandering set consist in only one fixed point.
A classical problem in dynamical systems is to measure the complexity of a map in terms of their orbits. One of the main tools we have to achieve this goal is entropy. However, many interesting families of dynamical systems have every element with zero-entropy. One of said families are the Morse-Smale diffeomorphisms. The first author and E. Pujals introduced the concept of generalized entropy that extends the classical notion of entropy. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces.
This study examines the tree models of mildly dissipative diffeomorphisms on the disk $\D$. These models are one-dimensional dynamical systems with ergodic aperiodic data as well as some properties of the original dynamics. The focus of this work is their quality as dynamical invariants in both the topological and ergodic sense.
We construct the complete set of orders of growth and we define on it the generalized entropy of a dynamical systems. With this object we provide a framework where we can study the separation of orbits of a map beyond the scope of exponential growth. We are going to show that this construction is particularly useful to study families of dynamical systems with vanishing entropy. Moreover, we are going to see that the space of orders of growth in which orbits are separated is wilder than expected. This is going to be achieved with different types of examples.
In this work we explore the problem of transitivity of volume-preserving skew-products endomorphisms of the n-torus. More specifically, we establish relationships between transitivity and the action induced by the skew-product in the fundamental group.
We consider the class of diffeomorphisms of a manifold that its differential keeps invariant a one-dimensional subbundle $E$. For that type of diffeomorphisms is naturally defined a one-parameter family called $E-$translation. We prove that if a diffeomorphisms in above mentioned class is conjugate to its $E-$translation and the conjugacy is at distance $α$-Hölder to the identity respect to the parameter and $α>1/2$, then the $E$-direction is hyperbolic. This theorem is also sharp as it is be discussed with some examples. We also deal with the continuously stable case in the Skew-Products context with one-dimensional fibers, requiring extra hypothesis along the fibers like either non-negative second derivative or negative Schwartzian.