Browsing by Subject "periodic points"

Now showing 1 - 7 of 7
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    Arithmetic properties of 3-cycles of quadratic maps over Q
    (Elsevier, 2022-11) Morton, Patrick; Raianu, Serban; Mathematical Sciences, School of Science
    It is shown that c = -29/16 is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map fc(x) = x2 + c has a rational periodic point of period 3. Several arithmetic conditions on the set of all such rational numbers c and the rational orbits of fc(x) are proved. A graph on the numerators of the rational 3-periodic points of maps fc is considered which reflects connections between solutions of norm equations from the cubic field of discriminant -23.
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    Periodic points of algebraic functions and Deuring’s class number formula
    (Springer, 2019) Morton, Patrick; Mathematical Sciences, School of Science
    The exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. This gives an interpretation of the case p = 2 of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations H−d(x) is given that is applicable for small periods. The pre-periodic points of ˆ F(z) in Q are also determined.
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    Periodic points of latitudinal maps of the m-dimensional sphere
    (AIMS, 2016-11) Graff, Grzegorz; Misiurewicz, Michał; Nowak-Przygodzki, Piotr; Department of Mathematical Sciences, School of Science
    Let f be a smooth self-map of the m-dimensional sphere Sm. Under the assumption that f preserves latitudinal foliations with the fibres S1, we estimate from below the number of fixed points of the iterates of f. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz.
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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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    Shub’s conjecture for smooth longitudinal maps of Sm
    (Taylor & Francis, 2018) Graff, Grzegorz; Misiurewicz, Michał; Nowak-Przygodzki, Piotr; Mathematical Sciences, School of Science
    Let f be a smooth map of the m-dimensional sphere Sm to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f, reduce Shub’s conjecture for longitudinal maps to a lower dimensional classical version, and prove the conjecture in case m=2 and in a weak form for m=3.
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    Solutions of diophantine equations as periodic points of p-adic algebraic functions, I
    (2016) Morton, Patrick; Department of Mathematical Sciences, School of Science
    Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields K = Q(√-d) with -d Ξ 1 (mod 8) are shown to be periodic points of a fixed algebraic function T(z) defined on the punctured disk 0 < |z|2 ≥ 1/2 of the maximal unramified, algebraic extension K2 of the 2-adic field Q2. All ring class fields of odd conductor over imaginary quadratic fields in which the prime p = 2 splits are shown to be generated by complex periodic points of the algebraic function T, and conversely, all but two of the periodic points of T generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime p in place of p = 2, where the case p = 3 has been previously proved by the author, and the case p = 5 will be handled in Part II.
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    Special α-limit sets
    (American Mathematical Society, 2020) Kolyada, Sergiǐ; Misiurewicz, Michał; Snoha, L’ubomír
    We investigate the notion of the special α-limit set of a point. For a continuous selfmap of a compact metric space, it is defined as the union of the sets of accumulation points over all backward branches of the map. The main question is whether a special α-limit set has to be closed. We show that it is not the case in general. It is unknown even whether a special α-limit set has to be Borel or at least analytic (it is in general an uncountable union of closed sets). However, we answer this question affirmatively for interval maps for which the set of all periodic points is closed. We also give examples showing how those sets may look like and we provide some conjectures and a problem.