Browsing by Author "Misiurewicz, Michal"

Now showing 1 - 7 of 7
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    Counting Preimages
    (Cambridge, 2017) Misiurewicz, Michal; Rodrigues, Ana; Department of Mathematical Sciences, School of Science
    For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.
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    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.
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    Periodic orbits of piecewise monotone maps
    (2018-04-23) Cosper, David; Misiurewicz, Michal
    Much is known about periodic orbits in dynamical systems of continuous interval maps. Of note is the theorem of Sharkovsky. In 1964 he proved that, for a continuous map $f$ on $\mathbb{R}$, the existence of periodic orbits of certain periods force the existence of periodic orbits of certain other periods. Unfortunately there is currently no analogue of this theorem for maps of $\mathbb{R}$ which are not continuous. Here we consider discontinuous interval maps of a particular variety, namely piecewise monotone interval maps. We observe how the presence of a given periodic orbit forces other periodic orbits, as well as the direct analogue of Sharkovsky's theorem in special families of piecewise monotone maps. We conclude by investigating the entropy of piecewise linear maps. Among particular one parameter families of piecewise linear maps, entropy remains constant even as the parameter varies. We provide a simple geometric explanation of this phenomenon known as entropy locking.
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    Periodic points of latitudinal sphere maps
    (Springer, 2014-12) Misiurewicz, Michal; Department of Mathematical Sciences, School of Science
    For the maps of the two-dimensional sphere into itself that preserve the latitude foliation and are differentiable at the poles, lower estimates of the number of fixed points for the maps and their iterates are obtained. Those estimates also show that the growth rate of the number of fixed points of the iterates is larger than or equal to the logarithm of the absolute value of the degree of the map.
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    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.
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    Symbolic dynamics for Lozi maps
    (IOP, 2016) Misiurewicz, Michal; Štimac, Sonja; Department of Mathematical Sciences, School of Science
    We study the family of Lozi maps ${{L}_{a,b}}:{{\mathbb{R}}^{2}}\to {{\mathbb{R}}^{2}}$ , ${{L}_{a,b}}(x,y)=(1+y-a|x|,bx)$ , and their strange attractors ${{ \Lambda }_{a,b}}$ . We introduce the set of kneading sequences for the Lozi map and prove that it determines the symbolic dynamics for that map. We also introduce two other equivalent approaches.
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    Topological entropy of Bunimovich stadium billiards
    (2020) Misiurewicz, Michal; Zhang, Hong-Kun; Mathematical Sciences, School of Science
    We estimate from below the topological entropy of the Bunimovich stadium billiards. We do it for long billiard tables, and find the limit of estimates as the length goes to infinity.