Consider $d$ disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated to the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of $C^1$ conjugation. We solve the inverse problem posed by Dennis Sullivan: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it.
Given a $C^{1+γ}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+γ}, C^\infty$ or $C^ω$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+γ}$ conjugacy class of $C$. The proof of this leads to the following rigidi…
Let $D$ be the set of $β\in (1, 2]$ such that $f_β$ is a symmetric tent map with finite critical orbit. For $β\in D$, by operating Denjoy like surgery on $f_β$, we constructed a $C^1$ unimodal map $\tilde{g}_β$ admitting a thick hyperbolic repelling invariant Cantor set which contains a wild Cantor attractor. The smoothness of $\tilde{g}_β$ is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. Furthermore, $D$ is dense in $(1, 2]$, and $\tilde{g}_β$ can not be $C^{1+α}$ because the hyperbolic repelling invaria…
We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, homeomorphisms that are topologically conjugate to g approximate f in the topology of uniform convergence if a trivial necessary condition on the periodic points holds. In particular, if f is a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point, then homeomorphisms on the Cantor set that are topologically conjugate to a subshift approximate f in the topology of uniform convergence. I…
In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequences of asymptotic length ratios are precisely those given by a positive Hölder continuous function $s$ (solenoid function) on the Cantor set $C$ of 2-adic integers satisfying a functional equation called the matching condition. The functional equation for the 2-adic integer Cantor set is $s(2x+1)=\frac{s(x)}{s(2x)}(1+\frac{1}{s(2x-1)})-1.$ We also present a one-to-one corr…
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hype…
Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known that $J_f$ is a Cantor sets if and only if there exists a positive integer $n>0$ such that $\bar{f^{-n}(U)}\subset U$ for some open topological disc $U$ containing no critical values. Let $n_f$ denote the minimal positive integer satisfying the above. The problem is whether $n_f=1$ or not. Let $S_d$ denote the shift locus of rational maps of degree $d$. We sh…
We study diagonal actions $φ:\mathbb{F}_2\curvearrowright\partial\mathbb{F}_2\times K$ on the Cantor set which are given by $φ_a=\partial_a\timesα,φ_b=\partial_b\timesβ$. Under some restrictions on $α,β$ we compute $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)$. As an application in the case of $α$ is Denjoy homeomorphism of the Cantor set and $β=id$ we will show that $C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2$ is Kirchberg algebra with $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)=(\mathbb{Z}^\infty,\mathbb{Z}^\infty)$. Also we will check that $C^*$-crosse…
We study renormalization of highly dissipative analytic three dimensional Hénon maps $$ F(x,y,z) = (f(x) - \varepsilon(x,y,z),\ x,\ δ(x,y,z)) $$ where $ \varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \varepsilon_{2d}(x,y) $. Under certain conditions, $ C^r $ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $ 2 \leq r < \infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ Hénon-like map. It also defines two dimensional embedded $ C^r $ …
In this paper I prove that for a polynomial of degree $d$ with a Cantor Julia set $J$, the Julia set can be understood as the simplest possible quotiont of the one sided shift space $Σ_d$ with dynamics given by the shift. Here simplest possible means that, the projection $π: Σ_d \Rightarrow J$ is as injective as possible.
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the two-sided shift.
In this article, we study countable sofic shifts of Cantor-Bendixson rank at most 2. We prove that their conjugacy problem is complete for GI, the complexity class of graph isomorphism, and that the existence problems of block maps, factor maps and embeddings are NP-complete.
We study sets of nontypical points under the map $f_β\mapsto βx $ mod 1, for non-integer $β$ and extend our results from [Färm, Persson, Schmeling, 2010] in several directions. In particular we prove that sets of points whose forward orbit avoid certain Cantor sets, and set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition $β>1.541$ found in [Färm, Persson, Schmeling, 2010] can be removed.
We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms between pseudogroups (in particular giving rise to equivariant maps at the level of spaces). As applications, we obtain explicit obstructions to the existence of embeddings between full groups in terms…
Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is an element of A^L that is X-admissible everywhere except for some small region of L which we call a `defect'. Such defects are analogous to `domain boundaries' in a crystalline solid. It has been empirically observed that these defects persist under iteration of F,…
We consider an infinite-to-one factor map from an irreducible shift of finite type X to a sofic shift Y. A compensation function relates equilibrium states on X to equilibrium states on Y. The p-Dini condition is given as a way of measuring the smoothness of a continuous function, with 1-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. We show that the relative equilibrium states of a 1-Dini function f over a fully supported invariant measure on Y are themselves fully supported, and have positive relative entro…
Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples sh…
Extendable Shift Maps and the Canonical Map on Generalized Countable Markov Shifts
Institute of Mathematics, Statistics and Computer Science (IME-USP), University of São Paulo（圣保罗大学数学、统计和计算机科学研究所）；Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教 Pontifícia 大学）；Institute of Physics (IFUSP), University of São Paulo（圣保罗大学物理研究所）
We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{η_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is Hölder continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, al…
