arXiv subjects
Publications and source records attributed to Henk Bruin.
We obtain limit theorems (Stable Laws and Central Limit Theorems, both Gaussian and non-Gaussian) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The proofs of the limit theorems for these flows are applications of corresponding theorems in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.
We study a one-parameter family of countably piecewise linear interval maps, which, although Markov, fail the `large image property'. This leads to conservative as well as dissipative behaviour for different maps in the family with respect to Lebesgue. We investigate the transition between these two types, and study the associated thermodynamic formalism, describing in detail the second order phase transitions (i.e. the pressure function is $C^1$ but not $C^2$ at the phase transition) that occur in transition to dissipativity. We also study the various natural definitions of pressure which arise here, computing these using elementary recurrence relations.
We prove that for a chainable continuum $X$ and every non-zigzag $x\in X$ there exists a planar embedding $ϕ:X\to ϕ(X)\subset\mathbb R^2$ such that $ϕ(x)$ is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings $ϕ,ψ:X \to \mathbb R^2$ are called strongly equivalent if $ϕ\circ ψ^{-1}: ψ(X) \to ϕ(X)$ can be extended to a homeomorphism of $\mathbb R^2$. We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.
A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval), and when it equals 0, namely, at finite direct bifurcations from the polynomial $z^2$, as well as limit points thereof.
A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.
The purpose of this paper is to establish limit laws for volume preserving almost Anosov flows on $3$-three manifolds having a neutral periodic of cubic saddle type. In the process, we derive estimates for the Dulac maps for cubic neutral saddles in planar vector fields.
We study the properties of folding points and endpoints of unimodal inverse limit spaces. We distinguish between non-end folding points and three types of end-points (flat, spiral and nasty) and give conditions for their existence and prevalence. Additionally, we give a characterisation of tent inverse limit spaces for which the set of folding points equals the set of endpoints.
We develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows. Previously, such results were restricted to the situation where there is a first return Poincaré map that is uniformly expanding and Markov. As illustrations of the method, we consider semiflows over nonMarkov Pomeau-Manneville intermittent maps with infinite measure, and we also obtain mixing rates for semiflows over Collet-Eckmann maps with nonintegrable roof function.
Assume that $(X,f)$ is a dynamical system and $ϕ:X \to [-\infty, \infty)$ is a potential such that the $f$-invariant measure $μ_ϕ$ equivalent to $ϕ$-conformal measure is infinite, but that there is an inducing scheme $F = f^τ$ with a finite measure $μ_{\barϕ}$ and polynomial tails $μ_{\barϕ}(τ\geq n) = O(n^{-β})$, $β\in (0,1)$. We give conditions under which the pressure of $f$ for a perturbed potential $ϕ+sψ$ relates to the pressure of the induced system as $P(ϕ+sψ) = (C P(\overline{ϕ+sψ}))^{1/β} (1+o(1))$, together with estimates for the $o(1)$-error term. This extends results from Sarig to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes for the Pomeau-Manneville map with potential $ϕ_t = - t\log f'$, as well as on the results by Bruin & Todd on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures $μ_{ϕ+sψ}$ as $s\to 0$ are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.
We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.
The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.
There is a natural connection between two types of recurrence law: hitting times to shrinking targets, and hitting times to a fixed target (usually seen as escape through a hole). We show that for systems which mix exponentially fast, one can move through a natural parameter space from one to the other. On the other hand, if the mixing is subexponential, there is a phase transition between the hitting times law and the escape law.
We consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching
We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of the tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ has topological entropy $\htop(h) = |R| \log s$, where $R \in \Z$ is such that $h$ and $σ^R$ are isotopic. Conclusions on the possible values of the entropy of homeomorphisms of the inverse limit space of a (renormalizable) quadratic map are drawn as well.
We prove that every self-homeomorphism on the inverse limit space of a quadratic map is isotopic to some power of the shift map.
Let $F$ be a (non-Markov) countably piecewise expanding interval map satisfying certain regularity conditions, and $\tildeŁ$ the corresponding transfer operator. We prove the Dolgopyat inequality for the twisted operator $\tildeŁ_s(v) = \tildeŁ_s(e^{sφ}v)$ acting on the space BV of functions of bounded variation, where $φ$ is a piecewise $C^1$ roof function.
We investigate matching for the family $T_α(x) = βx + α\pmod 1$, $α\in [0,1]$, for fixed $β> 1$. Matching refers to the property that there is an $n \in \mathbb N$ such that $T_α^n(0) = T_α^n(1)$. We show that for various Pisot numbers $β$, matching occurs on an open dense set of $α\in [0,1]$ and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent.
For point $x$ in the inverse limit space $X$ with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that $x$ is accessible. It follows that there are uncountably many non-equivalent planar embeddings of $X$.