Equi-topological entropy curves for skew tent maps in the square

We consider skew tent maps $T_{{\alpha}, {\beta}}(x)$ such that $({\alpha}, {\beta})\in[0,1]^{2}$ is the turning point of $T {_ {{\alpha}, {\beta}}}$, that is, $T_{{\alpha}, {\beta}}=\frac{{\beta}}{{\alpha}}x$ for $0\leq x \leq {\alpha}$ and $T_{{\alpha}, {\beta}}(x)=\frac{{\beta}}{1-{\alpha}}(1-x)$ for ${\alpha}<x\leq 1$. We denote by ${\underline{M}}=K({\alpha}, {\beta})$ the kneading sequence of $T_ {{\alpha}, {\beta}}$ and by $h({\alpha}, {\beta})$ its topological entropy. For a given kneading squence ${\underline{M}}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $({\alpha}, \varphi_{{\underline{M}}}({\alpha}))$ such that $K({\alpha}, {\varphi}_{{\underline{M}}}({\alpha}))= {\underline{M}}$. To study the behavior of these curves an auxiliary function ${\Theta}_{{\underline{M}}}({\alpha}, {\beta})$ is introduced. For this function ${\Theta}_{{\underline{M}}}({\alpha}, \varphi_{{\underline{M}}}({\alpha}))=0$, but it may happen that for some kneading sequences $\Theta_{{\underline{M}}}({\alpha}, {\beta})=0$ for some ${\beta}< \varphi_{{\underline{M}}}({\alpha})$ with $({\alpha}, {\beta})$ still in the interesting region. Using ${\Theta}_{{\underline{M}}}$ we show that the curves $({\alpha},\varphi_{{\underline{M}}}({\alpha}))$ hit the diagonal $\{({\beta}, {\beta}): 0.5< {\beta}<1 \}$ almost perpendicularly if $({\beta}, {\beta})$ is close to $(1,1)$. Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for ${\underline{M}}=RLLRC$ the curve $(\alpha, {\varphi}_{{\underline{M}}}({\alpha}))$ is not orthogonal to the diagonal. On the other hand, for ${\underline{M}}=RLC$ it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu

Ergodic averages with prime divisor weights in L-1

We show that omega (n) and Omega (n), the number of distinct prime factors of n and the number of distinct prime factors of n counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in L-1. That is, if g denotes one of these functions and S-g, K = Sigma(n infinity)1/S-g, K Sigma(K)(n=1)g(n)f(tau(n)x) = integral(x) f d mu for mu almost every x is an element of X. This answers a question raised by Cuny and Weber, who showed this result for L-p, p > 1

Equi-topological entropy curves for skew tent maps in the square

We consider skew tent maps Tα,β(x) such that (α,β)∈[0,1]2 is the turning point of Tα,β, that is, Tα,β=βαx for 0≤x≤α and Tα,β(x)=β1−α(1−x) for α<x≤1. We denote by M−−=K(α,β) the kneading sequence of Tα,β and by h(α,β) its topological entropy. For a given kneading squence M−− we consider equi-kneading, (or equi-topological entropy, or isentrope) curves (α,φM−−(α)) such that K(α,φM−−(α))=M−−. To study the behavior of these curves an auxiliary function ΘM−−(α,β) is introduced. For this function ΘM−−(α,φM−−(α))=0, but it may happen that for some kneading sequences ΘM−−(α,β)=0 for some β<φM−−(α) with (α,β) still in the interesting region. Using ΘM−− we show that the curves (α,φM−−(α)) hit the diagonal {(β,β):0.5<β<1} almost perpendicularly if (β,β) is close to (1,1). Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for M−−=RLLRC the curve (α,φM−−(α)) is not orthogonal to the diagonal. On the other hand, for M−−=RLC it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu

On sets where lip f is finite

Given a function f : R -> R, the so-called "little lip" function lip f is defined as follows:lip f(x) = lim inf(r SE arrow 0) sup(vertical bar x - y vertical bar <= r) vertical bar f(y) - f(x)vertical bar/r.We show that if f is continuous on R, then the set where lip f is infinite is a countable union of countable intersections of closed sets (that is, an F-sigma delta set). On the other hand, given a countable union E of closed sets, we construct a continuous function f such that lip f is infinite exactly on E. A further result is that, for a typical continuous function f on the real line, lip f vanishes almost everywhere

Multifractal spectrum and generic properties of functions monotone in several variables

International audienceWe study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let Ell be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f : [0, 1](d) -> R, we have that dim E (f)(h)= d - 1 + h for all h is an element of [0,1], and in addition, we obtain that the set E(f)(h) is empty as soon as h > 1. We also investigate the level set structure of such functions. (C) 2011 Elsevier Inc. All rights reserved

Singularity Spectrum of Generic alpha-Holder Regular Functions After Time Subordination

International audienceA question of Yves Meyer motivated the research concerning "time" subordinations of real functions. Denote by beta(alpha)(1) the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g is an element of beta(alpha)(1)one obtains a time subordination of g simply by considering the composite function Z=gau < f, where faa"(3):={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space epsilon = M X beta(alpha)(1) equipped with the product supremum metric is a complete metric space. In this paper for all alpha a[0,1) multifractal properties of gau < f are investigated for a generic (typical) element (f,g)aa"degrees (alpha) . In particular we determine the generic Holder singularity spectrum of gau < f