For arbitrary parabolically semihyperbolic generalized polynomial-like maps f , we prove that on a certain interval, which contains the interval (0; HD(J(f ))), the pressure function t 7! P( t log jf 0 j) is real-analytic. Our results generalize the work of Makarov and Smirnov in [3] and [7]. 1.

For S-unimodal maps f , we study equilibrium states maximizing the free energies F t () := h() t R log jf 0 jd and the pressure function P (t) := sup F t (). It is shown that if f is uniformly hyperbolic on periodic orbits, then P (t) is analytic for t 1. On the other hand, examples

x \Gamma a j a i \Gamma a j be the polynomial that is 1 at a i and 0 at all a j for j 6= i. Set Uniqueness follows the fact that over a field, every two degree d polynomials either agree on at most d points or agree on all points. 2 Lemma 3 Let jF j ? d. For every set of (d + 1) (point

Abstract. In this paper, we shall consider Privalov space Np0 (D) (p> 1) which consists of holomorphic functions f on the upper half plane D: = fz 2 C j Imz> 0g such that (log+ jf(z)j)p has a harmonic majorant on D. We shall give some properties of Np0 (D). 1.

Abstract. By de¯nition the asymptotic maximum principle (AMP) is valid for a class K of analytic functions in the unit disk D if, whenever f 2 K sup 0·µ<2 lim sup r!1 jf(reiµ)j = sup z2D jf(z)j: Let Kt (t ¸ 0) be the class of all f for which lim sup jzj!1 f(1 ¡ jzj)2 log jf(z)jg · t: Then AMP

analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form t log jF j, where F is the natural map associated with f closely related to the corresponding map introduced in [UZd]. It includes appropriately de ned topological pressure, Perron-Frobenius operators

mathematician has been exposed to in their education, but has only recently been seriously interpreted in this way, and that is Jensen’s theorem in complex analysis: This tells us that for a function f which is analytic on a closed disk of radius r, the average value of log jf(z)j on the boundary of the disk

in the river but were never detected in the inclined well. Throughout the study, the streambed demonstrated a range of reductions from 0.5 log to 4 log in the measured parameters. In this riverbank filtration system, the streambed is a significant barrier against infiltration of contaminants. KEY TERMS

derivative Sf = f 000 f 0 \Gamma 3 2 ( f 00 f 0 ) 2 . For such a map set '(x) := log jf 0 (x)j and 'n (x) := '(x) + '(fx) + \Delta \Delta \Delta + '(f n\Gamma1 x). Let \Pi n = fx 2 [0; 1] : f n (x) = xg and define for t 2 R the zeta function i t (z) = exp 1 X n

logarithmic Sobolev inequality with constant C if Ent i f 2 j C Z jrf j 2 d (1) for all smooth enough functions f where Ent i f 2 j = Z f 2 log f 2 d \Gamma Z f 2 d log Z f 2 d : Let us recall two consequences of this property for . First, for every r 0, and every Lipschitz