A Class of Möbius Iterated Function Systems

Ends of iterated function systems

Guided by classical concepts, we define the notion of \emph{ends} of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points are then linked to idempotent maps. A commutative diagram illustrates the natural relationships between the infinite walks in a semigroup and components of an attractor in more detail. We show in particular that, if an iterated function system is one-ended, the associated attractor is connected, and ask whether every connected attractor (fractal) conversely …

Asymptotics for a class of iterated random cubic operators

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single maps, starting with a generic initial condition where in particular all genetic types occur with positive frequency, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur. We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particula…

Hartman-Grobman Theorem for IFS

In this paper, for iterated function systems, we define the classic concept of the dynamical systems: topological conjugacy of diffeomorphisms. We generalize the Hartman-Grobman theorem for one dimensional iterated function systems on R. Also, we introduce the basic concept of structural stability for an iterated function system and so we investigate the necessary condition for structural stability of an iterated function system on R.

无限视界洛伦兹气体的广义迭代对数律

Generalized law of iterated logarithm for the Lorentz gas with infinite horizon

本文为超扩散相依过程建立了广义迭代对数律，并应用于无限视界洛伦兹气体，扩展了经典极限定理的适用范围。

Ergodicity of iterated function systems via minimality on the hyper spaces

We give a sufficient condition for the ergodicity of the Lebesgue measure for an iterated function system of diffeomorphisms. This is done via the induced iterated function system on the space of continuum (which is called hyper-space). We introduce a notion of minimality for induced IFSs which implies that the Lebesgue measure is ergodic for the original IFS. Here, to beginning, the required regularity is $C^1$. However, it is proven that the $C^1$-regularity is a redundant condition to prove ergodicity with respect to the class of quasi-invariant measures. As a consequence of mentioned resul…

Weakly contractive iterated function systems and beyond: A manual

We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ("chaos game"), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contrac…

Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems

We study existence and uniqueness of the fixed points solutions of a large class of non-linear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry-Mather theory, Sinai-Ruelle-Bowen measures, fat solenoidal attractors, a…

Real Projective Iterated Function Systems

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined …

位置依赖的幂等迭代函数系统的不变测度

Invariant measures for place dependent idempotent iterated function systems

该研究针对紧度量空间上的位置依赖幂等迭代函数系统，利用Mañé势、Aubry集等动力系统方法刻画了不变幂等概率的密度，并给出一类模糊迭代函数系统吸引子的新公式。

关于全纯迭代函数系统的边界动力学的注记

A note on the boundary dynamics of holomorphic iterated function systems

本文研究全纯自映射的单位圆盘迭代函数系统的边界动力学，提出一个充分条件，将内部动力行为扩展到边界，推广了Bourdon等人对经典Denjoy-Wolff定理的扩展。

扩张迭代函数系的相对变分原理

A Relative Variational Principle for Expanding Iterated Function Systems

本文针对一类扩张迭代函数系，证明了相对变分原理，利用非平稳Ruelle-Perron-Frobenius定理建立了符号基遍历不变测度下边缘熵与诱导斜积纤维平均拓扑熵的关系。

A class of continua that are not attractors of any IFS

This paper presents a sufficient condition for a continuum in $R^n$ to be embeddable in $R^n$ in such a way that its image is not an attractor of any iterated function system. An example of a continuum in $R^2$ that is not an attractor of any weakly contracting iterated function system is also given.

Random countable iterated function systems with overlaps and applications

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the i…

A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the l…

Using Iterated Function Systems to Reveal Biases in the Distribution of Prime Numbers

Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes, having specified arithmetic or congruence properties. Some of the phenomena shown in our computations relate to the recent, groundbreaking work of Lemke Oliver and Soundararajan on biases between consecutive primes. We do not have asymptotics to explain our results, but provide graphs, data, and det…

Directed porosity on conformal iterated function systems and weak convergence of singular integrals

The aim of the present paper is twofold. We study directed porosity in connection with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures $μ$ and kernels $K$, converge weakly in some dense subspaces of $L^2(μ)$ when the support of $μ$ belongs to a broad family of sets. This class contains many fractal sets like CIFS's limit sets.

Thermodynamic formalism for invariant measures in iterated function systems with overlaps

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for iterated function systems with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $ν= π_*μ$ is a self-conformal measure, then $HD(ν) < \frac{h(μ)}{|χ(μ)|}$ i…

Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy

We describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS). We discuss questions related to the Markov property and we present a definition of entropy which is induced by a QIFS. This definition is a natural generalization of the Shannon-Kolmogorov entropy from Ergodic Theory. This definition is different from the one in the paper "A Thermodynamic Formalism for density matrices in Quantum Information" by the same authors.

On the dimensions of attractors of random self-similar graph directed iterated function systems

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separati…