Conformal Iterated Function Systems Research Articles

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

Transversal family of non-autonomous conformal iterated function systems

Transversal family of non-autonomous conformal iterated function systems

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

We consider small perturbations of a conformal iterated function system produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai–Ruelle–Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications to the dimension theory of continued fraction Cantor sets include strengthening Hensley’s asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $$\le N$$ ; as well as its counterpart for reals whose partial quotients are all $$\ge N$$ due to Good from 1941.

DIMENSIONS OF MEASURES, DEGREES, AND FOLDING ENTROPY IN DYNAMICS

In this survey, we present some methods in the dynamics and dimension theory for invariant measures of hyperbolic endomorphisms (smooth non-invertible maps), and for conformal iterated function systems with overlaps. For endomorphisms, we recall the notion of asymptotic degree of an equilibrium measure, which is shown to be related to the folding entropy; this degree is then applied to dimension estimates. For finite iterated function systems, we present the notion of overlap number of a measure, which is related to the folding entropy of a lift transformation, and also give some examples when it can be computed or estimated. We apply overlap numbers to prove the exact dimensionality of invariant measures, and to obtain a geometric formula for their dimension. Then, for countable conformal iterated function systems with overlaps, the projections of ergodic measures are shown to be exact dimensional, and we give a dimension formula. Relations with ergodic number theory, continued fractions, and random dynamical systems are also presented.

Modified shrinking target problems on self-conformal sets

In this paper, we investigate a modified version of the shrinking target problem on self-conformal sets, which unifies the shrinking target problems and quantitative recurrence properties. Let J be a self-conformal set generated by a conformal iterated function system satisfying the open set condition, and let T:J→J be the expanding map induced by the left shift. We will study the size of the following set:R(f,φ):={x∈J:|Tnx−f(x)|<φ(n)for infinitely manyn∈N}, where f:J→J is a Lipschitz function and φ:N→R+ is a positive function defined on N. The Hausdorff dimension and zero-one law on the μ-measure of R(f,φ) are completely obtained, where μ stands for the natural self-conformal measure supported on J.

Geometry of Measures in Random Systems with Complete Connections

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and place-dependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps \(\mathcal S\), and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system \(\mathcal S\).

Dimensions of slices through the Sierpiński gasket

We discuss the intersections of the d-dimensional Sierpiński gasket with -dimensional hyperplanes in a particular fixed direction. We give the values of the Hausdorff dimension, the lower box dimension, the upper box dimension, and the packing dimension of each slice and determine the criterion for which the Hausdorff dimension of the slice takes Marstrand's value. In order to investigate the slices, we improve the theory of non-autonomous conformal iterated function systems.

On the dimension spectra of infinite conformal iterated function systems

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions – (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect – which resolve questions posed by Chousionis, Leykekhman and Urbański, and goes on to provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.

The Hausdorff dimension spectrum of Renyi-like continued fractions

In this paper, we look at a family of Renyi-like continued fraction expansions and the associated conformal iterated function systems. We show that for every k≥2, every such associated system has full Hausdorff dimension spectrum. We construct two examples of iterated function systems that do not have full Hausdorff dimension spectrum. One is a subsystem of any given Renyi-like system, while the second consists of similarities.

Dynamics of hyperbolic correspondences

AbstractThis paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

On Spectral Asymptotics of the Sturm–Liouville Problem with Self-Conformal Singular Weight

Under study is the spectral asymptotics of the Sturm–Liouville problem with a singular self-conformal weight measure. We assume that the conformal iterated function system generating the weight measure satisfies a stronger version of the bounded distortion property. The power exponent of the main term of the eigenvalue counting function asymptotics is obtained under the assumption. This generalizes the result by Fujita in the case of self-similar (self-affine) measures.

Multifractal formalism for generalised local dimension spectra of Gibbs measures on the real line

We extend the multifractal formalism for the local dimension spectrum of a Gibbs measure μ supported on the attractor Λ of a conformal iterated functions system on the real line. Namely, for α∈R, we establish the multifractal formalism for the Hausdorff dimension of the set of x∈Λ for which the μ-measure of a ball of radius rn centred at x obeys a power law rnα, for a sequence rn→0. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.

Shrinking targets for non-autonomous systems

In the present work we establish a Bowen-type formula for the Hausdorff dimension of shrinking target sets for non-autonomous conformal iterated function systems in arbitrary dimensions and satisfying certain conditions. In the case of dimension 1 we also investigate non-linear perturbations of linear systems and obtain sufficient conditions under which the perturbed systems satisfy the conditions in our hypotheses.

On Spectral Asymptotics of the Sturm–Liouville Problem with Self-Conformal Singular Weight with Strong Bounded Distortion Property

Spectral asymptotics of the Neumann problem for the Sturm–Liouville equation with singular selfconformal weight measure is considered under the assumption of a stronger version of the bounded distortion property for the conformal iterated function system corresponding to the weight measure. The power exponent of the main term of the eigenvalue counting function asymptotics is obtained. This generalizes a result obtained by T. Fujita (1985) in the case of self-similar (selfaffine) measure.

Thin annuli property and exponential distribution of return times for weakly Markov systems

We deal with the problem of asymptotic distribution of first return times to shrinking balls under iteration generated by a large general class of dynamical systems called weakly Markov. Our ultimate main result is that these distributions converge to the exponential law when the balls shrink to points. We apply this result to many classes of smooth dynamical systems that include conformal iterated function systems, rational functions on the Riemann sphere $\widehat{\mathbb C}$, and transcendental meromorphic functions on the complex plane $\mathbb{C}$. We also apply them to expanding repellers and holomorphic endomorphisms of complex projective spaces. One of the key ingredients in our approach is to solve the well known, in this field of mathematics, problem of appropriately estimating the measures of, suitably defined, large class of geometric annuli. We successfully do it. This problem is, in the existing literature, differently referred to by different authors; we call it the Thick Thin Annuli Property. Having this property established, we prove that for non--conformal systems the aforementioned distributions converge to the exponential one along sets of radii whose relative Lebesgue measure converges fast to one. But this is not all. In the context of conformal iterated function systems, we establish the Full Thin Annuli Property, which gives the same estimates for all radii. ln this way, we solve a long standing problem. As a result, we prove that the convergence to the exponential law holds along all radii for essentially all conformal iterated function systems and, with the help of the techniques of first return maps, for all aforementioned conformal dynamical systems.

广义复连分数共形迭代系统的Hausdorff维数

广义复连分数共形迭代系统的Hausdorff维数

The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions

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 belongs to the boundary of the parameter space.

Hereditarily non uniformly perfect non-autonomous Julia sets

&lt;p style='text-indent:20px;'&gt;Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [&lt;xref ref-type="bibr" rid="b19"&gt;19&lt;/xref&gt;] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [&lt;xref ref-type="bibr" rid="b15"&gt;15&lt;/xref&gt;].&lt;/p&gt;

Hyperbolic Lifts and Estimates for Overlap Numbers

We first compute topological overlap numbers in several concrete cases. In particular is obtained a class of Bernoulli convolutions systems which asymptotically are irrational-to-1 on their limit sets. Next, for general conformal iterated function systems and for a class of potentials $$\psi $$ , we prove an estimate on the box dimension of a measure $$\nu _\psi $$ on the limit set, associated to an equilibrium measure $$\mu _\psi $$ on the respective hyperbolic lift. Also we study the structure of the families of conditional measures of $$\mu _\psi $$ with respect to various fiber partitions, and find relations between them.