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

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h})$ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions $f$ and all $h\geq 0$. We prove that for typical/generic (in the sense of Baire) continuous convex functions $f:[0,1]^{d}\to \mathbb{R}$, one has $\dim E_f(h) =d-2+h$ for all $h\in[1,2],$ and in addition, we obtain that the set $E_f({h} )$ is empty if $h\in (0,1)\cup (1,+\infty)$. Also, when $f$ is typical, the boundary of $[0,1]^{d}$ belongs to $E_{f}({0})$

Measures and functions with prescribed homogeneous multifractal spectrum

In this paper we construct measures supported in $[0,1]$ with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of $[0,1]$ has the same multifractal spectrum as the whole measure. The spectra $f$ that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of $[0,1]$ and satisfy $f(h)\leq h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure: the support of its spectrum within $[0,1]$ must be an interval. This result is a sort of Darboux theorem for multifractal spectra of measures. This result is optimal, since we construct a HM measure with spectrum supported by $[0,1] \cup {2}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.Comment: 34 pages, 6 figure

Generic Birkhoff Spectra

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and ${\sigma}$ is the one-sided shift. The Birkhoff spectrum $\displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},$ where $\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_{f}( {\alpha})$ is a bounded and closed interval $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$ and $S_{f}( {\alpha})$ on $L_{f}$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C( {\Omega})$ in the sense of Baire category. For a dense set in $C( {\Omega})$ the spectrum is not continuous on ${\mathbb {R}}$, though for the generic $f\in C( {\Omega})$ the spectrum is continuous on ${\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of $L_{f}$. We give an example of a function which has continuous $S_{f}$ on ${\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of $L_{f}$. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $f$ and $g$ are close in $C( {\Omega})$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.Comment: Revised version after the referee's repor

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

Monotone and convex restrictions of continuous functions

Suppose that f belongs to a suitably defined complete metric space C-alpha of Milder alpha-functions defined on [0,1]. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets A subset of [0,1] such that f vertical bar(A) is monotone, or convex/concave. Some of our results are about generic functions in C-alpha like the following one: we prove that for a generic f is an element of C-1(alpha)[0, 1], 0 < alpha < 2 for any A subset of [0,1] such that f vertical bar(A) is convex, or concave we have dim(H) A <= dim(M) A <= max{0, alpha - 1}. On the other hand we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for 1 < alpha <= 2 for any f is an element of C-alpha [0,1] there is always a set A subset of [0,1] such that dim(H) A = alpha - 1 and f vertical bar(A) is convex, or concave on A. (C) 2017 Elsevier Inc. All rights reserved

Tensor Products of AC* Charges and AC Radon Measures Are Not Always AC* Charges

AbstractIn this note we give an example of an AC* charge, F, on R and an absolutely continuous Radon measure μ on R such that F⊗μ is not an AC* charge on R2