Browsing by Subject "Complex Dynamics"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    The Dynamics of Semigroups of Contraction Similarities on the Plane
    (2019-08) Silvestri, Stefano; Perez, Rodrigo; Geller, William; Misiurewicz, Michal; Roeder, Roland K.
    Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.
  • Thumbnail Image
    Item
    Some Connections Between Complex Dynamics and Statistical Mechanics
    (2020-05) Chio, Ivan; Roeder, Roland K. W.; Misiurewicz, Michal; Perez, Rodrigo A.; Yattselev, Maxim L.
    Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.