Henk Bruin's Research Interests

Henk Bruin

Absolutely continuous invariant probabilities (acips) of interval maps

The long-term behaviour of chaotic dynamical systems can be described statistically by means of invariant probability measures m. The frequency that typical orbits spend in a specified subset of the phase space tends to the measure of that subset (Birkhoff's Ergodic Theorem, provided m is ergodic). "Typical" is related to the measure, but if the measure is absolutely continuous with respect to Lebesgue measure (m is an acip), then the above holds for Lebesgue almost all points. For smooth maps of the interval, finding acips is non-trivial, and is in general connected to the growth rate of the derivative along the orbits of the critical points. A recurring theme in my research is the (non)existence of acips when this growth rate is very small, see e.g.

Infinite measured systems

There are many dynamical systems that do not preserve a finite measure that is absolutely continuous with respect to the reference measure (such as Lebesgue for interval maps). Examples of such maps are the Manneville-Pomeau map, but also certain (quadratic) interval maps. In most cases, sigma-finite measures exist (although not always), and in all cases the statistical properties of such measures are strikingly different form what we understand in the probability measure case.

Statistical properties of non-uniformly hyperbolic dynamical systems

Ergodic theory of chaotic dynamical systems aims at describing the system in terms of statistical laws that are frequently coming from stochastic processes. Laws that we would like to establish include the Central Limit Theorem, decay of correlations (rates of mixing), return time statistics to small sets, Bernoulli property (i.e. isomorphism to a Bernoulli process). Operator theory and inducing techniques play an important role in establishing such properties.

Equilibrium states

Equilibrium states are invariant measures that balance their potential energy with their entropy content. This approach to selecting invariant mirrors ideas from statistical mechanics. The thermodynamic pressure is the supremum, taken over all invariant measures, of the sum of potential energy and entropy.
Equilibrium states of Holder potentials of uniformly hyperbolic systems are fairly well understood; this theory goes back to Bowen, Ruelle and Sinai in the mid 1970s. Since ca. 2000 the field has regained interest as the knowledge of nonuniformly hyperbolic systems increased with the advent of new tools, such as (Young) towers and Markov extensions. <\br> Our research shows existence and uniqueness of equilibrium states for interval maps f and the natural potential -t log|Df|. Also we show that the thermodynamic pressure where f is an interval map. Also we show that the thermodynamic pressure depends analytically on the parameter t near t=1. and equilibrium states, i.e. measures that maximize the sum of there entropy nd the integral w.r.t. a specified potential function:

Attractors of chaotic dynamical systems

The Rauzy fractal is the factor of
a  certain wild attractor Attractors of one-dimensional systems can be more interesting than simply stable periodic orbits. There are ways in which a Cantor set A can attract Lebesgue a.e. point. One is the "solenoidal attractor", when the map is infinitely renormalisable, the other is the "wild attractor", in which case the second Baire category set of points is not attracted to A. Thus A is only an attractor in a measure theoretic, but not in a topological sense. The following paper deal with the properties of such attractors:

Symbolic dynamics

Bratteli diagrams

Symbolic descriptions are very common in symbolic dynamics, leading to several classes of subshifts, such as subshifts of finite type, sofic systems, Toeplitz shifts, substitution shifts, etc. In my research, especially the latter plays a role in describing the dynamics of maps restricted to certain minimal Cantor sets. In fact, any minimal system on the Cantor set has descriptions in terms of substitutions shifts, enumeration scales as well as adic shifts on Bratteli diagrams.

Topological dynamics of interval maps

poodle fractal is a factor of the attractor of the tribonacci map The combinatorial and topological properties of interval maps and maps of the complex plane have an impact on their metric properties. But also they are also interesting on their own accord, as they provided new approaches to, e.g. the structure of minimal Cantor systems within interval maps.

Topological entropy of interval maps

Entropy is not a monotone
				     function of a single critical
				     value for the cubic family Topological entropy is a measure of the complexity of a map. One outstanding question is how this depends on the parameters in parametrised families of interval maps. In the main conjecture from the 1980s in this area, namely that entropy depends monotonically on the parameter in the logistic family was solved by results of Milnor and Thurston, Sullivan, and Douady and Hubbard. All these results use complex methods, and a purely real proof is still unknown. In fact, monotonicity fails for certain non-logistic families. Some results of mine in this area, including the proof of monotonicity of entropy (joint with Sebastian van Strien) for multimodal polynomials are the following.

Inverse limit spaces

a Henon attractor

An inverse limit is a topological construction, which, briefly said, consists of all backward orbits of a dynamical system equipped with a suitable topology. If the dynamical system is a unimodal (say quadratic) map, this resulting space resembles to some extend the well-known Henon attractor (see left). self-asymptotic arc-composant I view the study of unimodal inverse limit spaces as step towards studying the topology of Henon attractors. Whereas Henon attractors are at least as complicated as unimodal inverse limit spaces (with a single bonding map), unimodal inverse limit spaces are sufficiently complicated themselves to leave us with many questions. Ingram's Conjecturm, i.e., the question of whether inverse limit spaces of non-conjugate bonding maps are always non-homeomorphic was solved in 2009 in a joint paper by Marcy Barge, Sonja Stimac and me.

Complex dynamics

The Julia set of z^2+i

Hubbard tree Julia sets of maps on the complex plane are a well-known source of fractals, and especially for quadratic polynomials, there is a lot of literature, software and books about their Julia sets and the Mandelbrot set. In joint work with Dierk Schleicher, we describe the underlying combinatorial structure of complex quadratic polynomials, their symbolic dynamics and Hubbard trees, external rays both in dynamic and parameter space, and algorithms to connect them all.

Piesewise isometries

overflow Piecewise isometries appear in applications ranging from digital data processing (Sigma-Delta-modulators), polygonal billiards and queuing theory. Their dynamics is usually a mixture of (quasi)periodic and chaotic motion, where it should be noted that the chaos is due o the discontinuities in the system, rather than to positive Lyapunov exponents or positive entropy.
Except for a few special cases, which use number-theoretic peculiarities, the behaviour of piecewise isometries poses many unanswered questions. Already in dimension 1, there are piecewise isometries with very interesting properties, e.g. they can possess attractors of a multifractal nature, and carrying multiple ergodic invariant measures.



Last modified: June 17 2011