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Orbit Representations from Linear mod 1 Transformations
Ramos, Carlos Correia; Martins, Nuno; Pinto, Paulo R.We show that every point x0 2 [0; 1] carries a representation of a C -algebra that encodes the orbit structure of the linear mod 1 interval map f ; (x) = x + . Such C -algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map f ; . Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and onl...
Iteration of quadratic maps on matrix algebras
Baptista, Alexandra Nascimento; Ramos, Carlos Correia; Martins, NunoWe study the iteration of a quadratic family in the algebra of 2 × 2 real matrices, parameterized by a matrix C. We analyze and classify the existing cycles (periodic orbits) and their dependence on the parameter matrix. We discuss how new dynamical phenomena occur as a consequence of the noncommutativity of the matrix product. In particular, we show that the commutator of the initial condition with parameter m...
Dynamics on certain sets of stochastic matrices
Ramos, Carlos Correia; Martins, Nuno; Alexandra, Nascimento BaptistaWe study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stocha...
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