Dynamics Of Polynomials Research Articles

On the dynamics of real polynomials on the plane

We analyze the dynamics of the map f(z) = z 2 − s z ̄ where s ∈ C is a constant and z ∈ C is a variable. For some values of s, we can have invariant measures of density with respect to the two-dimensional Lebesgue measure. For other values of s we can have fractal repellers or the X-trange attractor. This problem is related to a Triple Point Phase Transition Model (Potts Model).

The dynamics of complex polynomials and automorphisms of the shift

The dynamics of complex polynomials and automorphisms of the shift

Marker automorphisms of the one-sided d-shift

AbstractWe identify a set of generators for the automorphism group of the one-sided d-shift. For the 3-shift, this set of generators has an application to the dynamics of cubic polynomials.

Dynamics of real polynomials on the plane and triple point phase transition

We will consider the Generalized Chebyshev polynomials on the plane and show the existence of three equilibrium probabilities for the pressure associated with a distinguished value of an external parameter. The phenomena shows an intriguing analogy with the Potts Model of Statistical Mechanics. The main feature of this model is to explain the sudden magnetization of a Ferromagnetic system at a certain value of the temperature. In the Potts model three equilibrium measures coexist at the transition value. In our case two of these equilibrium probabilities have magnetic (or anti-ferromagnetic) nature, and the other one is not of magnetic nature. This phenomena can be classified as a triple point phase transition. For temperatures below this distinguished value, there exist just two equilibrium measures, namely the two magnetic and anti-ferromagnetic ones mentioned above Our main theorem is presented in §2, after we explain some properties of the dynamics of the Generalized Chebyshev Polynomials. The mathematical model presented here has a strong analogy with the phenomena of triple point transition in a semi-infinite one-dimensional spin-lattice N; with four spin-components in each sitte of the lattice. One should also suppose the existence of an anisotropy in the lattice in such way some of the spin-components (or directions) are not favorable. Some results related to Yang-Lee zeros are presented in the end of the paper. We refer the reader to the conclusion at the end of the paper for more comments about the above considerations.