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We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\\ell $ . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when $\\ell $ tends to $\\infty $ . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

We study distributions of the harmonic and Lebesgue measures on the boundary of the connectedness locus M d for uni-critical polynomials z d + c and dynamical properties which are typical with respect to the harmonic measure. One result is a conformal similarity between M d and the corresponding Julia set for almost every point of M d with respect to the harmonic measure. It is also shown that the harmonic measure is supported on Lebesgue density points of the complement of M d which are not accessible from outside within John angles and at which the boundary of M d ‘spirals’ infinitely often in both directions. In the case of quadratic polynomials, we prove that every parameter c from the boundary of the Mandelbrot set M that is recurrent under iterates of z 2 + c but not infinitely tunable is a Lebesgue weak density point of the complement of M . A direct consequence of our theory is the Yoccoz local connectivity theorem. A new technical ingredient in the paper is the method of “amplification” which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding Julia set. The pointwise conformality of the similarity map is derived through inducing techniques and a classical TWB-theory.

We study the problem of the existence of wild attractors for critical circle coverings with Fibonacci dynamics. This is known to be related to the drift for the corresponding fixed points of renormalization. The fixed point depends only on the order of the critical point $\\ell$ and its drift is a number $\\unicode[STIX]{x1D717}(\\ell )$ which is finite for each finite $\\ell$ . We show that the limit $\\unicode[STIX]{x1D717}(\\infty ):=\\lim _{\\ell \\rightarrow \\infty }\\unicode[STIX]{x1D717}(\\ell )$ exists and is finite. The finiteness of the limit is in a sharp contrast with the case of Fibonacci unimodal maps. Furthermore, $\\unicode[STIX]{x1D717}(\\infty )$ is expressed as a contour integral in terms of the limit of the fixed points of renormalization when $\\ell \\rightarrow \\infty$ . There is a certain paradox here, since this dynamical limit is a circle homeomorphism with the golden mean rotation number whose own drift is $\\infty$ for topological reasons.

Minocycline (MINO), a classic antibiotic, may have psychotropic activity related to the modulation of the tryptophan-kynurenine pathway. In this study, we investigated the effects of MINO on (1) memory and anxiety behaviors, (2) the modulation of brain levels of amyloid precursor protein (APP) and 2,3-indoleamine dioxygenase (IDO1) levels, and (3) peripheral inflammatory markers in a streptozotocin (STZ)-induced rat model of sporadic Alzheimer’s disease (sAD). After repeated treatment with a dose of 35 mg/kg MINO for seven consecutive days, male Wistar rats with sAD showed (1) improvements in early (29 days after injection, probe test) reference memory (decreased latency to reach the platform, increased time in the critical quadrant of the Morris water maze) and anxiety disorders (increased time in the open arms of the elevated plus maze; increased exploration and entrances in the center of the white–light illuminated open field) 45–46 and 90–91 days after STZ injection; (2) reduced APP and IDO1 levels in the hippocampus and prefrontal cortex; and (3) induction of anti-inflammatory response in blood (increased TCD4+ lymphocyte number and interleukin-10 production). This suggests that MINO, due to its anti-inflammatory action, improves memory and anxiety behavior related to sAD, indicating its neuroprotective and psychotropic properties.

Drug-induced immunosuppression may underline increased hypothalamic–pituitary–adrenal axis response to stress observed following chronic psychostimulant treatment. However, the consequences of random amphetamine (AMPH) treatment, withdrawal and AMPH challenge after withdrawal on the peripheral immunity and systemic corticosterone response are unknown. In this study, the total blood and spleen leukocyte, lymphocyte, T, B, NK, TCD4 + /TCD8 + cell numbers and ratio, pro-inflammatory interferon gamma (IFN-γ), and anti-inflammatory interleukin-4 (IL-4) production, and plasma corticosterone concentration in Wistar rats were investigated after: chronic, random AMPH/SAL treatment alone (20 injections in 60 days, 1 mg/kg b.w., i.p.), AMPH/SAL withdrawal (for 20 consecutive days after random AMPH/SAL exposure) or AMPH/SAL challenge after withdrawal (single injection after the AMPH/SAL withdrawal phase). The results showed blood and spleen leukopenia, lymphopenia, lower blood production of IFN-ɤ, and increased plasma corticosterone concentration after the AMPH treatment, which were more pronounced in the AMPH after withdrawal group. In contrast, an increased number of blood NK cells and production of IL-4 after chronic, random AMPH treatment alone, were found. Blood AMPH-induced leukopenia and lymphopenia were due to decreased total number of T, B lymphocytes and, at least in part, of granulocytes and monocytes. Moreover, decreases in the number of blood TCD4 + and TCD8 + lymphocytes both in the AMPH chronic alone and withdrawal phases, were found. The major findings of this study are that AMPH treatment after the long-term withdrawal from previous random AMPH exposure, accelerates the drug-induced immunosuppressive and systemic corticosterone responses, suggesting prolonged immunosuppressive effects and an increase in incidence of infectious diseases. Graphical Abstract Prolonged peripheral immunosuppressive responses as consequences of random amphetamine…The results indicate that the chronic and random AMPH exposure alone and the acute (single injection) challenge of the drug after the withdrawal phase induced long-term immunosuppressive effects, which were similar to those occurring during the stress response, and sensitized the peripheral immunosuppressive and corticosterone responses of the rat to the disinhibitory effects of this stressor.

Individual differences in responsiveness to environmental factors, including stress reactivity and anxiety levels, which differ between high (HR) and low (LR) responders to novelty, might be risk factors for development of memory and anxiety disorders in sporadic Alzheimer’s disease (sAD). In the present study, we investigated whether behavioral characteristics of the HR and LR rats, influence the progression of sAD (neuroinflammation, β-amyloid peptide, behavioral activity related to memory (Morris water maze) and anxiety (elevated plus maze, white and illuminated open field test) in streptozotocin (STZ)-induced neuroinflammation as a model of early pathophysiological alterations in sAD. Early (45 days) in disease progression, there was a more severe impairment of reference memory and higher levels of anxiety in HRs compared with LRs. Behavioral depression in HRs was associated with higher expression of β-amyloid deposits, particularly in the NAcS, and activation of microglia (CD68+ cells) in the hypothalamus, as opposed to less inflammation in the hippocampus, particularly in CA1, compared with LRs in late (90 days) sAD progression. Our findings suggest that rats with higher behavioral activity and increased responsivity to stressors show more rapid progression of disease and anxiety disorders compared with low responders to novelty in the STZ-induced sAD model.

We classify the measure theoretic attractors of general C3unimodal maps with quadratic critical points. The main ingredient is the decay of geometry.

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial z−p(z)¯z-\\overline {p(z)}, deg⁡p=n>1\\deg p = n > 1, has at most 3n−23n-2 complex zeros.

Intracerebroventricular-(ICV)-streptozotocin-(STZ)-induced neuroinflammation is a model of Alzheimer's disease (AD) compatible with the inflammation hypothesis of ageing (\"inflammaging\" state). Previously, we observed age-dependent (young vs aged) dimethyl fumarate (DMF)-induced anti-inflammatory and neuroprotective effects in the brain along with improvement in cognitive functions in rats with the ICV-STZ-induced model of AD. To evaluate whether DMF reduces neuroinflammation based on the peripheral inflammatory response inhibition, we determined peripheral inflammatory mediators in young and aged rats with the ICV-STZ-induced AD pathology following DMF therapy. Young (4-month-old) and aged (22-month-old) rats were fed with 0.4% DMF rat chow for 21 consecutive days after ICV-STZ (3 mg/ventricle) injections. After behavioral testing, blood and spleens were collected to determine the numbers of leukocytes (WBC), lymphocytes and their subpopulations, haematological parameters, the concanavalin (Con)-A-induced production and plasma concentration of interferon (IFN)-γ, interleukin (IL)-6, IL-10 and corticosterone (COR). Age-dependent anti-inflammatory effect of the DMF treatment in rats with ICV-STZ injections manifested as decreased peripheral WBC and lymphocyte numbers, including TCD3 CD4 CD8 , TCD3 CD4 CD8 , B (CD45RA ) and NK (161a ), in aged rats. Furthermore, DMF lowered the blood and spleen lymphocyte production of pro-inflammatory IFN-γ and IL-6 in young and aged rats, whereas it enhanced the plasma level of anti-inflammatory IL-10 and lymphocyte's ability to produce it in aged rats only. In parallel to changes in peripheral WBC numbers in the model of AD, DMF decreased the red blood cell number, haemoglobin concentration, haematocrit and mean platelet volume in aged, but not young, rats. In contrast to controls, DMF did not influence the COR response in STZ groups. Besides preventing neuroinflammation, DMF acts on the pro-/anti-inflammatory balance in the periphery and causes an anti-inflammatory shift in T lymphocytes which could contribute to DMF's therapeutic effects in the ICV-STZ-induced model of AD, in particular, in aged rats.