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Points with maximal Birkhoff average oscillation

Jinjun Li, Min Wu (2016)

Czechoslovak Mathematical Journal

Let f : X → X be a continuous map with the specification property on a compact metric space X . We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally...

Separation conditions on controlled Moran constructions

Antti Käenmäki, Markku Vilppolainen (2008)

Fundamenta Mathematicae

It is well known that the open set condition and the positivity of the t-dimensional Hausdorff measure are equivalent on self-similar sets, where t is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions in this respect.

Sets of nondifferentiability for conjugacies between expanding interval maps

Thomas Jordan, Marc Kesseböhmer, Mark Pollicott, Bernd O. Stratmann (2009)

Fundamenta Mathematicae

We study differentiability of topological conjugacies between expanding piecewise C 1 + ϵ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum....

Subadditive Pressure for IFS with Triangular Maps

Balázs Bárány (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate properties of the zero of the subadditive pressure which is a most important tool to estimate the Hausdorff dimension of the attractor of a non-conformal iterated function system (IFS). Our result is a generalization of the main results of Miao and Falconer [Fractals 15 (2007)] and Manning and Simon [Nonlinearity 20 (2007)].

The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition

Michael Keane, K. Károly, Boris Solomyak (2003)

Fundamenta Mathematicae

Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.

The growth rate and dimension theory of beta-expansions

Simon Baker (2012)

Fundamenta Mathematicae

In a recent paper of Feng and Sidorov they show that for β ∈ (1,(1+√5)/2) the set of β-expansions grows exponentially for every x ∈ (0,1/(β-1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.

The recurrence dimension for piecewise monotonic maps of the interval

Franz Hofbauer (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [ 0 , 1 ] , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...

Trajectories of polynomial vector fields and ascending chains of polynomial ideals

Dmitri Novikov, Sergei Yakovenko (1999)

Annales de l'institut Fourier

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ℝ n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension.The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned...

Upper Estimate of Concentration and Thin Dimensions of Measures

H. Gacki, A. Lasota, J. Myjak (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

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