38 research outputs found
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
Fast and slow points of Birkhoff sums
International audienceWe investigate the growth rate of the Birkhoff sums ,where is a continuous function with zero mean defined on the unit circle and isa ``typical'' element of . The answer depends on the meaning given to the word ``typical''. Part of the work will be done in a more general context
Typical Borel measures on satisfy a multifractal formalism
In this article, we prove that in the Baire category sense, measures
supported by the unit cube of typically satisfy a multifractal
formalism. To achieve this, we compute explicitly the multifractal spectrum of
such typical measures . This spectrum appears to be linear with slope 1,
starting from 0 at exponent 0, ending at dimension at exponent , and it
indeed coincides with the Legendre transform of the -spectrum associated
with typical measures .Comment: 17 pages. To appear in Nonlinearit
Topological Hausdorff dimension and level sets of generic continuous functions on fractals
In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dim H K and dim tH K denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely sup{ dimHf- 1(y):y∈R}= dimtHK-1 for the generic f ∈ C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dim H f -1(y) = dim tH K - 1 for the generic f ∈ C(K) and the generic y ∈ f(K). The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic f ∈ C(K) and the generic y ∈ f(K) we have dim H f -1(y) = dim tH K - 1. We also generalize a result of B. Kirchheim by showing that if K is self-similar then for the generic f ∈ C(K) for every y∈intf(K) we have dim H f -1(y) = dim tH K - 1. Finally, we prove that the graph of the generic f ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K. © 2012 Elsevier Ltd. All rights reserved
Multifractal properties of convex hulls of typical continuous functions
We study the singularity (multifractal) spectrum of the convex hull of the
typical/generic continuous functions defined on . We denote by
the set of points at which has a pointwise H\"older exponent equal to . Let
be the convex hull of the graph of , the concave function on the top
of is denoted by and denotes the convex function on
the bottom of .
We show that there is a dense subset such that for the following properties are
satisfied. For the functions and
coincide only on a set of zero Hausdorff dimension, the functions are continuously differentiable on , equals the boundary of ,
, and if
Type 1 and 2 sets for series of translates of functions
Suppose Lambda is a discrete infinite set of nonnegative real numbers. We say that Lambda is type 1 if the series s(x)=Sigma lambda is an element of Lambda f(x+lambda) satisfies a zero-one law. This means that for any non-negative measurable f:R ->[0,+infinity) either the convergence set C(f,Lambda)={x:s(x)<+infinity}=R modulo sets of Lebesgue zero, or its complement the divergence set D(f,Lambda)={x:s(x)=+infinity}=R modulo sets of measure zero. If Lambda is not type 1 we say that Lambda is type 2.The exact characterization of type 1 and type 2 sets is not known. In this paper we continue our study of the properties of type 1 and 2 sets. We discuss sub and supersets of type 1 and 2 sets and give a complete and simple characterization of a subclass of dyadic type 1 sets. We discuss the existence of type 1 sets containing infinitely many elements independent over the rationals. Finally, we consider unions and Minkowski sums of type 1 and 2 sets
Random constructions for translates of non-negative functions
Suppose A is a discrete infinite set of nonnegative real numbers. We say that A is type 2 if the series s(x) = Sigma lambda Lambda f (x + lambda) does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" f : R -> [0,+ infinity) such that both the convergence set C(f, Lambda) ={x : s(x) < + infinity} and its complement the divergence set D (f, Lambda) = {x : s(x) = +infinity} are of positive Lebesgue measure. If Lambda is not type 2 we say that A is type 1. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set A if we randomly delete its elements. Motivated by results concerning weighted sums Sigma c(n)f(nx)and the Khinchin conjecture, we also discuss some results about weighted sum
Pointwise convergence of Birkhoff averages for global observables
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in
infinite ergodic theory is trivial; it states that for any
infinite-measure-preserving ergodic system the Birkhoff average of every
integrable function is almost everywhere zero. Nor does a different rescaling
of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In
this paper we give a version of Birkhoff's theorem for conservative, ergodic,
infinite-measure-preserving dynamical systems where instead of integrable
functions we use certain elements of , which we generically call
global observables. Our main theorem applies to general systems but requires an
hypothesis of "approximate partial averaging" on the observables. The idea
behind the result, however, applies to more general situations, as we show with
an example. Finally, by means of counterexamples and numerical simulations, we
discuss the question of finding the optimal class of observables for which a
Birkhoff theorem holds for infinite-measure-preserving systems.Comment: Final version. 33 pages, 10 figure