136 research outputs found
Equi-topological entropy curves for skew tent maps in the square
We consider skew tent maps such that is the turning point of , that
is, for and for . We denote by the
kneading sequence of and by its
topological entropy. For a given kneading squence we
consider equi-kneading, (or equi-topological entropy, or isentrope) curves
such that . To study the behavior
of these curves an auxiliary function is introduced. For this function ,
but it may happen that for some kneading sequences
for some with still in the
interesting region. Using we show that the curves
hit the diagonal almost perpendicularly if is
close to . 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 the curve is not orthogonal to the diagonal. On
the other hand, for 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 . Let be the set of points at which
has a pointwise exponent equal to . We first obtain general upper bounds
for the Hausdorff dimension of these sets , for all convex functions
and all . We prove that for typical/generic (in the sense of
Baire) continuous convex functions , one has for all and in addition, we obtain that the set is empty if . Also, when is typical,
the boundary of belongs to
Measures and functions with prescribed homogeneous multifractal spectrum
In this paper we construct measures supported in with prescribed
multifractal spectrum. Moreover, these measures are homogeneously multifractal
(HM, for short), in the sense that their restriction on any subinterval of
has the same multifractal spectrum as the whole measure. The spectra
that we are able to prescribe are suprema of a countable set of step
functions supported by subintervals of and satisfy for all
. We also find a surprising constraint on the multifractal spectrum
of a HM measure: the support of its spectrum within 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 . Using wavelet theory, we also build HM functions
with prescribed multifractal spectrum.Comment: 34 pages, 6 figure
Generic Birkhoff Spectra
Suppose that and is the
one-sided shift. The Birkhoff spectrum where is the
Hausdorff dimension. It is well-known that the support of is
a bounded and closed interval
and on is concave and upper semicontinuous. We are
interested in possible shapes/properties of the spectrum, especially for
generic/typical in the sense of Baire category. For a dense
set in the spectrum is not continuous on ,
though for the generic the spectrum is continuous on , but has infinite one-sided derivatives at the endpoints of
. We give an example of a function which has continuous on , but with finite one-sided derivatives at the endpoints of
. The spectrum of this function can be as close as possible to a
"minimal spectrum". We use that if two functions and are close in then and are close on 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
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