Total disconnectedness of Julia sets of random quadratic polynomials

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

Characteristics of Patients Showing Discrepancy Between Bruch’s Membrane Opening-Minimum Rim Width and Peripapillary Retinal Nerve Fiber Layer Thickness

Background: To investigate clinical characteristics of patients showing discrepancy between Bruch’s membrane opening minimum rim width (BMO-MRW) and peripapillary retinal nerve fiber layer (RNFL) thickness. Correlation with the visual field (VF) was also inspected. Methods: In this prospective, cross-sectional study, 106 eyes (106 subjects) showing normal BMO-MRW classification but abnormal RNFL classification were included. All patients underwent confocal scanning laser ophthalmoscopy, spectral-domain optical coherence tomography, and standard automated perimetry. Results: Clinical characteristics were as follows: mean age: 52.79 ± 14.75 years; spherical equivalent (SE), −2.52 ± 3.48 diopter (D); SE < −5.0 D, 34 (32.1%) eyes; large disc (>2.43 mm2), 40.6%; small disc (<1.63 mm2), 12.5%; VF index, 96.72 ± 9.58%; mean deviation, −1.74 ± 3.61 dB; β-peripapillary atrophy (PPA), 96.2%; γ-PPA, 75.5%. Majority (86.1%) of these cases demonstrated normal (71.3%) or borderline (14.9%) on VF. Temporal and nasal RNFL showed significant differences among disc size subgroups (all p < 0.05). Nasal RNFL was significantly thicker in a large disc group than other subgroups. Temporal, superotemporal, inferotemporal, inferonasal RNFL, and superior RNFL peak location showed significant differences (all p < 0.05) among SE subgroups. Temporal RNFL was significantly thicker in the high myopia group than other subgroups. Conclusions: Temporalization of RNFL peaks in myopia and nasalization of RNFL peaks in large disc that display abnormal classifications might show normal classification of BMO-MRW. These findings of discrepancy between classifications should be considered in the diagnosis of early glaucoma.

Ocular findings in patients with novel A/H1N1 infection: an observational cross-sectional study

Background: The systemic findings of novel A (H1N1) influenza have been well documented but data on ocular lesions is scarce. We report the systemic and ophthalmoscopic findings of 14 patients with proven H1N1 infection. Methods: An observational non-interventional retrospective study. During a period of nine years, 14 patients (six female and eight males), were referred for an ophthalmic evaluation. All patients underwent a detailed systemic and ocular evaluation. Results: Three patients (21.4 %) showed ocular lesions in the form of a unilateral intraretinal hemorrhage. Of these, one patient (7.1%) had a large disc hemorrhage and an area of retinal whitening, consistent with ischemia in the macular area. Conclusions: Physicians dealing with H1N1 infection should be aware of these findings and may include an ocular evaluation as part of their protocol.