Some techniques and results in the renormalization theory of real and complex dynamical systems are summarized. The construction of the induced Markov map of [−1, 1] from a Feigenbaum-like map is presented. We show that this induced Markov map has bounded geometry. We discuss some property of infinitely renormalizable quadratic polynomials and show that the Julia set of an infinitely renormaliz...

We propose using discrete dynamical systems to model non-Markovian processes. This implies a whole hierarchy of dynamical systems with dimensionality increasing proportional to the memory. Specific long-range memory effects are investigated for a quadratic, a cubic and a quartic map. Markovian processes transform the quadratic map into a normal form of the logistic equation; the H6non map corre...

Our general goal is to extract frequent patterns in a database of images, where images are modelled by combinatorial maps. To do so, we define two nD combinatorial map signatures : the first one has a quadratic space complexity and may be used to decide of isomorphism with a new map in linear time whereas the second one has a linear space complexity and may be used to decide of isomorphism in q...

In this paper, we address the problem of computing a canonical representation of an n-dimensional combinatorial map. To do so, we define two combinatorial map signatures: the first one has a quadratic space complexity and may be used to decide of isomorphism with a new map in linear time whereas the second one has a linear space complexity and may be used to decide of isomorphism in quadratic t...

We prove that almost every real quadratic map is either regular or Collet-Eckmann. This implies exponential rates of mixing of the last renormalization of the map. We also show that the recurrence of the critical orbit is polynomial in a full measure set. It follows from a result of Baladi and Viana that almost every quadratic map is stochastically stable in the strong sense. Combined with meth...

We propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real Hénon family, we prove the existence of many regions of hyperbolic parameters in the parameter plane of the family.

We consider perturbations of the complex quadratic map $ z \to z^2 +c$ and corresponding changes in their quasi-Mandelbrot sets. Depending on particular perturbation, visual forms of quasi-Mandelbrot set changes either sharply (when the perturbation reaches some critical value) or continuously. In the latter case we have a smooth transition from the classical form of the set to some forms, cons...

It has been observed that for most classes of planar maps, the number of maps of size n grows asymptotically like c · n−5/2γn, for suitable positive constants c and γ. It has also been observed that, if dk is the limit probability that the root vertex in a random map has degree k, then again for most classes of maps the tail of the distribution is asymptotically of the form dk ∼ c·kq as k →∞, f...