In this paper, we investigate parameter families of iterated function systems and continuity. Specifically, if we have a set of iterated function systems that depend continuously on a parameter, which properties of the invariant sets will vary continuously? We show here that the invariant sets will vary continuously as a function of the parameter. Furthermore, we give some sufficient conditions under which the Hausdorff dimension will depend continuously on the parameter. Lastly, we give a counter-example where the dimension is not a continuous function of the parameter.
We consider iterated function systems on the interval with random perturbation. Let $Y_ε$ be uniformly distributed in $[1- ε, 1 + ε]$ and let $f_i \in C^{1+α}$ be contractions with fixpoints $a_i$. We consider the iterated function system $\{Y_εf_i + a_i (1 - Y_ε) \}_{i=1}^n$, were each of the maps are chosen with probability $p_i$. It is shown that the invariant density is in $L^2$ and the $L^2$-norm does not grow faster than $1/\sqrtε$, as $ε$ vanishes.
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained from the most natural covering system. We also prove that for Lebesgue typical parameters, the 1-dimensional Lebesgue measure of the underlying attractor is positive, if this number is bigger than 1 and all the contraction ratios are positive.
This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the "$λ$-topology". This classification theorem allows us to generalize some geometric results from the paper ("Lambda-Topology vs. Pointwise Topology", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clea…
This paper examines thresholds for certain properties of the attractor of a general one-parameter affine family of iterated functions systems. As the parameter increases, the iterated function system becomes less contractive, and the attractor evolves. Thresholds are studied for the following properties: the existence of an attractor, the connectivity of the attractor, and the existence of non-empty interior of the attractor. Also discussed are transition phenomena between existence and non-existence of an attractor.
We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.
We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges in distribution to the normal distribution.
We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + λ_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $λ_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors $y_1,y_2,...$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $χ= E[\log(λY)]$ be the Lyapunov exponent. Assuming that $χ< 0$, we obtain a family of conditional meas…
This paper discusses, certain algebraic, analytic, and topological results on partial iterated function systems($IFS_p$'s). Also, the article proves the Collage theorem for partial iterated function systems. Further, it provides a method to address the points in the attractor of a partial iterated function system and obtain results related to the address of points in the attractor. The completeness of the partial metric space of contractions with a fixed contractivity factor is proved, under suitable conditions. Also, it demonstrates the continuity of the map that associates each contraction i…
It is known that the Principal Poincaré Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincaré Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.
We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures. In addition we consider the multifractal analysis for countable families of potentials. We obtain a conditional variational principle for the level sets.
The affinity dimension is a number associated to an iterated function system of affine maps, which is fundamental in the study of the fractal dimensions of self-affine sets. De-Jun Feng and the author recently solved a folklore open problem, by proving that the affinity dimension is a continuous function of the defining maps. The proof also yields the continuity of a topological pressure arising in the study of random matrix products. I survey the definition, motivation and main properties of the affinity dimension and the associated SVF topological pressure, and give a proof of their continui…
Numerical solutions are presented for a family of finite angular momentum orbits with three equal masses which connects the classical circular Lagrange orbit with the recently discovered planar figure eight orbit. Each member of this family is a periodic orbit in a frame rotating around the horizontal symmetry axis of the figure eight orbit. Numerical evidence is given that this family is a continuous function of the angular rotation frequency. Similar numerical solutions are also found for n>3 equal masses, where n is an odd integer. Finite angular momentum orbits also have been obtained for …
We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.
Miculescu and Mihail in 2008 introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS. Instead of families of selfmaps of a metric space $X$, they considered families of mappings defined on finite Cartesian product $X^m$. It turned out that a great part of the classical Hutchinson--Barnsley theory has natural counterpart in this GIFSs' case. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ and obtained versions of the Hutchinson--Barnsley …
Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain families. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for iterated preimages of rational functions and relate this conjecture to other well-known conjectures in arithmetic dynamics.
We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.
We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^λ_j\}$ on the real line. Given a family of parameter dependent measures $\{μ_λ\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $Π^λ$. The main novelty of our work is that the measures $μ_λ$ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of a…
This is the second part of our study of the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on ${\Bbb R}^d$. In the first part we proved that the upper box-counting dimension of the attractor of any $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parametrized families of $C^1$ IFSs, and show that these upper bounds give actually the dimensions for…
We investigate whether the Hutchinson operator associated with the iterated function system (IFS) is continuous. It clarifies several partial results scattered across recent literature. While the main example for IFS with strict attractor was provided by the family of contractions (the so-called hyperbolic system), the accent was put on ensuring that various contractivity-like conditions are preserved when the Hutchinson operator is induced, unless very recently it was discovered that strict attractors are quite often present for a large class of noncontractive maps, namely projective maps. Th…
Let $μ$ be a planar self-similar measure with similarity dimension exceeding $1$, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections $π_{e\sharp}(μ)$ are absolutely continuous for all $e \in S^{1} \setminus E$, where the exceptional set $E$ has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend previous work of Shmerkin and Solomyak fro…
We investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discrete dynamical system in which each time step corresponds to the application of one of a finite collection of maps. The maps, which represent distinct dynamical regimes, may act in some pre-determined sequence or may be applied in random order. An algorithm is developed to detect the sequence of regime switches under the assumption of continuity. This method is tested on a simple IFS and applied to an experimental computer performance data set. This methodology has a wide range of potential uses:…
We study non-autonomous conformal iterated function systems, with finite or countably infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a "Moran set construction".) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen's formula. We a…
We construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit Interval, and this yields an analogue of the classical Sturm-L…
Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of $K_\mathcal{A}$ is given by the minimum of 1 and the critical exponent, under the assumption that $\mathcal{A}$ satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semig…
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and …
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of that entropy with respect to the parameter. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of t…
The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated …
Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by Bárány and Käenmäki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters…
The sub-additive pressure function $P(s)$ for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution $s_0$ to $P(s_0)=1$, were introduced by K. Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affini…
We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1…
Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is normal. This is deduced from a result about the multipliers of iterated polynomials.
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems our results apply to include those arising from Bernoulli convol…
In 2013 Balka and Máthé showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS, in which they considered families of mappings defined on finite Cartesian product $X^m$ with values in $X$. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ endowed with supremum metric. In the paper we show that in Euclidean spaces a typ…
We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrized families of self-similar sets and measures in the complex plane, extending earlier results.
We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish…
Miculescu and Mihail in 2008 introduced the concept of a \emph{generalized iterated function system} (GIFS in~short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space~$X$, GIFSs consist of maps defined on a finite Cartesian {$m$-th power} $X^m$ with values in $X$ (in such a case we say that a GIFS is \emph{of order} $m$). It turned out that a great part of the classical {Hutchinson theory} has natural counterpart in this GIFSs' framework. On the other hand, there are known only few examples of~fractal sets which are generated by G…
We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.
We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belon…
We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continui…
We consider families of dynamics that can be described in terms of Perron-Frobenius operators with exponential mixing properties. For piecewise C^2 expanding interval maps we rigorously prove continuity properties of the drift J(l) and of the diffusion coefficient D(l) under parameter variation. Our main result is that D(l) has a modulus of continuity of order O(|dl||log|dl|)^2), i.e. D(l) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are verif…
We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family. We also introduce a notion of "first pointwise expansive power" of an eventually expansive interval map. We prove that for every map …
