We investigate in this work some situations where it is possible to estimate or determine the upper and the lower $q$-generalized fractal dimensions $D^{\pm}_μ(q)$, $q\in\mathbb{R}$, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem~\cite{Young} for the generalized fractal dimensions of the Bowen-Margulis measure associated with a $C^{1+α}$-Axiom A system over a two-dimensional compact Riemannian manifold $M$. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy.
Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures such that $D_μ^+(q)=0$ ($q\ge 1$), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\in [0,1)$, $D^{+}_μ(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric.
Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in~\cite{Sigmund1974} for Lipschitz transformations which satisfy the specification property.
We describe a class of fractal attractors induced by β-shifts. We use a coding by these shifts to show that the systems are mixing with topological entropy log βand have an ergodic measure of full entropy. Moreover we determine the Hausdorff dimension of the attractor.
The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given f…
For simplest colored branching processes we prove an analog to the McMillan theorem and calculate Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting the role of Shannon's entropy is played by the Kullback--Leibler divergence and the Hausdorff dimensions are computed by means of the so-called Billingsley--Kullback entropy, defined in the paper.
We prove an estimation of the Kolmogorov $ε$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.
Let $0< λ< μ<1$ and $λ+μ>1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{λ,μ}$ of the IFS $\{(λx,μy), (μx+1-μ, λy+1-λ)\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{λ, μ}$ has a box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if $λμ\ge 2^{-1/6}$, then $A_{λ,μ}$ has a non-empty interior.
One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention probability is assigned with which we retain the cube. The interior of the discarded cubes contain no points from the Fractal percolation set. In the retained ones we repeat the process ad infinitum. The set that remains after infinitely many steps is the Fractal percolation set. The homogeneous case is when all of these probabilities are the same. Recently, there …
This paper is in the form of an essay. It defines fractal tops and code space structures associated with set-attractors of hyperbolic iterated function systems (IFSs). The fractal top of an IFS is associated with a certain shift invariant subspace of code space, whence the entropy of the IFS, and of its set-attractor, may be defined. Given any ordered pair of hyperbolic IFSs, each with the same number of maps, there is a natural transformation, constructed with the aid of fractal tops, whose domain is the attractor A of the first IFS and whose range is contained in the attractor B of the secon…
In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered Følner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some Følner sequence) equals its topological entropy. This answers questions by Zheng and Chen (Israel Journal of Mathematics 212 (2016), 895-911) and Simpson (Theory Comput. Syst. 56 (2015), 527-543).
Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph. Using the first result, we investigate the change of volume entropy when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm of calculating the persistent volume entropy of graphs.
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a…
In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non-compact) locally compact Abelian groups that contain Meyer sets.
Cell-based models provide a helpful approach for simulating complex systems that exhibit adaptive, resilient qualities, such as cancer. Their focus on individual cell interactions makes them a particularly appropriate strategy to study the effects of cancer therapies, which often are designed to disrupt single-cell dynamics. In this work, we also propose them as viable methods for studying the time evolution of cancer imaging biomarkers (IBM). We propose a cellular automata model for tumor growth and three different therapies: chemotherapy, radiotherapy, and immunotherapy, following well-estab…
The subject of this note is the mixed Katugampola fractional integral of a bivariate function defined on a rectangular region in the Cartesian plane. This is a natural extension of the Katugampola fractional integral of a univariate function - a concept well-received in the recent literature on fractional calculus and its applications. It is shown that the mixed Katugampola fractional integral of a prescribed bivariate function preserves properties such as boundedness, continuity and bounded variation of the function. Furthermore, we estimate fractal dimension of the graph of the mixed Katugam…
We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy. Applications to Markov chains, Gibbs states and the stochastic scrabble are given.
Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.
We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Ha…
In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.
