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Henk Bruin

Publications and source records attributed to Henk Bruin.

At least 55 records · Page 3Linked to original sources
arXiv2016

A renewal scheme for non-uniformly hyperbolic semiflows

We investigate a renewal scheme for non-uniformly hyperbolic semiflows that closely resembles the renewal scheme developed in the discrete time case, in order to obtain sharp estimates for the correlation function. Also, the involved observables are supported on a flow-box of unbounded length. The present abstract setting does not require the use of Markov structure. However, the classes of examples covered here are rather restrictive. In these examples, it is easier to exploit the full force of the method and get optimal results for observables supported on finite length flow-boxes.

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arXiv2016

Constant slope maps and the Vere-Jones classification

We study continuous countably piecewise monotone interval maps, and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase our conditions in the terminology of the Vere-Jones classification of infinite matrices.

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arXiv2015

The Core Ingram Conjecture for non-recurrent critical points

We study inverse limit spaces of tent maps, and the Ingram Conjecture, which states that the inverse limit spaces of tent maps with different slopes are non-homeomorphic. When the tent map is restricted to its core, so there is no ray compactifying on the inverse limit space, this result is referred to as the Core Ingram Conjecture. We prove the Core Ingram Conjecture when the critical point is non-recurrent and not preperiodic.

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arXiv2015

Upper and lower bounds for the correlation function via inducing with general return times

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure preserving) non-Markov intervals maps with an indifferent fixed point.

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arXiv2015

Wild attractors and thermodynamic formalism

Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family $f_λ$ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential $ϕ_t = -t\log|f'_λ|$, there is a unique phase transition at some $t_1 \le 1$, and the pressure $P(ϕ_t)$ is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic $C^\infty$ curve (with all derivatives equal to 0 at $t_1 < 1$) at the emergence of a wild attractor, whereas the phase transition at $t_1 = 1$ can be of any finite order for those $λ$ for which $f_λ$ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of $ω(c)$.

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arXiv2014

On "observable" Li-Yorke tuples for interval maps

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect to Lebesgue), then the $d$-tuples have Lebesgue full measure, but if $T$ preserves an infinite absolutely continuous measure, the situation becomes more interesting. Taking the family of Manneville-Pomeau maps as example, we show that for any $d \ge 2$, it is possible that the set of Li-Yorke $d$-tuples has full Lebesgue measure, but the set of Li-Yorke $d+1$-tuples has zero Lebesgue measure.

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arXiv2013

Monotonicity of entropy for real multimodal maps

In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will prove the general case.

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arXiv2013

Renormalization, Freezing Phase Transitions and Fibonacci Quasicrystals

We examine the renormalization operator determined by the Fibonacci substitution. We exhibit a fixed point and determine its stable leaf (under iteration of the operator). Then, we study the thermodynamic formalism for po- tentials in this stable leaf, and prove they have a freezing phase transition, with ground state supported on the attracting quasi-crystal associated to the Fibonacci substitution

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arXiv2013

Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture

We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits.

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arXiv2013

On the structure of isentropes of polynomial maps

The structure of isentropes (i.e. level sets of constant topological entropy) including the monotonicity of entropy, has been studied for polynomial interval maps since the 1980s. We show that isentropes of multimodal polynomial families need not be locally connected and that entropy does in general not depend monotonically on a single critical value.

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arXiv2012

Bernoulli measure of complex admissible kneading sequences

Iterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the \emph{kneading sequence} which is an infinite sequence over $\{0,\1\}$. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{0,\1\}$.

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arXiv2012

Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\CR$ acting on the space of potentials are such that the pressure function $γ\mapsto \CP(-γV)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of $\CR$, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

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arXiv2011

The topological entropy of Banach spaces

We investigate some properties of (universal) Banach spaces of real functions in the context of topological entropy. Among other things, we show that any subspace of $C([0,1])$ which is isometrically isomorphic to $\ell_1$ contains a functions with infinite topological entropy. Also, for any $t \in [0, \infty]$, we construct a (one-dimensional) Banach space in which any nonzero function has topological entropy equal to $t$.

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arXiv2010

On isotopy and unimodal inverse limit spaces

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $σ^R : K_s \to K_s$.

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arXiv2009

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.

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