2 results
On the stochastic bifurcations regarding random iterations of polynomials of the form
$z^{2} + c_{n}$
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 26 February 2024, pp. 3358-3384
- Print publication:
- November 2024
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- Article
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In this paper, we consider random iterations of polynomial maps
$z^{2} + c_{n}$, where 
$c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter 
$c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.
Sign Changes of the Liouville Function on Quadratics
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 2 / 01 June 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 251-257
- Print publication:
- 01 June 2013
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Let
$\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla
conjectured that*
$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$for any polynomial
$f\left( x \right)$ with integer coefficients which is not of form 
$bg{{\left( x \right)}^{2}}$.
