## Henk Bruin's Research Interests

### 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.
• H. Bruin, Topological conditions for the existence of invariant measures for unimodal maps, Ergod. Th. and Dyn. Sys. 14 (1994) 433-451. [.pdf]
• H. Bruin, The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps, Ergod. Th.and Dyn. Sys. 18 (1998) 555-565. [.pdf]
• H. Bruin, S. van Strien, Existence of acips for multimodal maps, in Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday (2001) [.pdf]
• H. Bruin, W. Shen, S. van Strien, Invariant measures exist without a growth condition, Commun. Math. Phys. 241 (2003) 287-306 [.pdf]
The original publication is available at [http://www.springerlink.com]
• H. Bruin, W. Shen, S. van Strien, Existence of unique SRB-measures is typical for unimodal families, Annales Scientifiques de l'ENS, 4e serie, 39 (2006) 381-414. [.pdf]
• H. Bruin, J. Rivera-Letelier, W. Shen, S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Inventiones Mathematicae 172 (2008), 509-533, [.pdf] The original publication is available at [http://www.springerlink.com].
• H. Bruin, M. Todd, Complex maps without invariant densities, Nonlinearity 19 (2006) 2929-2945. Preprint (Version October 2006). [.pdf]

### 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.

• H. Bruin, J. Hawkins, Examples of expanding C1 maps having no sigma-finite invariant measure equivalent to Lebesgue, Isr. J. Math. 108 (1998) 83-107, [.pdf]
• J. Al-Khal, H. Bruin, M. Jakobson, New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure, Discrete and Continuous Dynamical Systems, 22 no. 1-2, (2008) 35 - 61, see [here] for journal's version. Earlier preprint version in [.pdf]
• H. Bruin, M. Jakobson, and appendix by Genadi Levin, New examples of topologically equivalent S-unimodal maps with different metric properties, Contemp. Math. 469 2008 119-139. [.pdf]
• H. Bruin, M. Todd, Complex maps without invariant densities, Nonlinearity 19 (2006) 2929-2945. Preprint (Version October 2006). [.pdf]
• H. Bruin, M. Nicol, D. Terhesiu, On Young towers associated with infinite measure preserving transformations, Stoch. and Dynamics, 9 (2009), 635 - 655, in [.pdf], see also World Scientific Publishing Company

### 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.
• H. Bruin, S. Luzzatto, S. van Strien, Decay of correlations in one-dimensional dynamics, Preprint (1999) and (2001), Ann. Sci. Ec. Norm. Sup. 36 (2003) 621-646, [.pdf]
• H. Bruin, Dimensions of recurrence times and minimal subshifts, Conference Proceedings: from Chrystal to Chaos, Luminy (2000), [.pdf]
• H. Bruin, Mark Holland, Ian Melbourne, Subexponential decay of correlations for compact group extension of nonuniformly expanding systems, Ergod. Th. & Dyn. Sys. 25 (2005) 1719-1738, [.pdf]
• H. Bruin, B. Saussol, S. Troubetzkoy, S. Vaienti, Return time statistics via inducing, Ergod. Th. Dyn. Sys. 23 (2003) 991-1013, [.pdf]
• H. Bruin, S. Vaienti, Return time statistics for unimodal maps, Fund. Math. 176 (2003) 77--94, [.pdf]
• H. Bruin, Mark Holland, Matt Nicol, Livsic regularity for Markov Systems, Ergod. Th. & Dyn. Sys. 25 (2005) 1739-1766, [.pdf]

### 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:

• H. Bruin, G. Keller, Equilibrium states for unimodal maps, Ergod. Th. and Dyn. Sys. 18 (1998) 765-789, [.pdf]
• H. Bruin, M. Todd, Equilibrium states for interval maps: potentials with sup φ - inf φ < h top(f), Commun. Math. Phys. 283 (2008) 579-611. DOI: 10.1007/s00220-008-0596-0. The original publication is available at this Springer site. Preprint version of February 2008, [.pdf]
• Henk Bruin, Mike Todd, Equilibrium states for interval maps: the potential -t log |Df|. Ann. Sci. Ecol Norm. Sup. 42 2009, 559-600. Preprint version of January 2009: [.pdf]
• H. Bruin, M. Todd Transience and thermodynamic formalism for infinitely branched interval maps Preprint 2011 [.pdf]. Journal of the London Mathematical Society 86 2012, 171-194 DOI: 10.1112/jlms/jdr081
• H. Bruin, M. Todd Wild attractors and thermodynamic formalism, Preprint 2012 [.pdf]
• (with Renaud Leplaideur) Renormalization, thermodynamic formalism and quasi-crystals in subshifts. [.pdf] Commun. Math. Phys. 321 (2013), 209-247. DOI: 10.1007/s00220-012-1651-4

### Attractors of chaotic dynamical systems

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:

• H. Bruin, G. Keller, T. Nowicki, S. van Strien, Wild Cantor attractors exist, Annals of Math. 143 (1996) 97-130, [.pdf]
• H. Bruin, G. Keller, M. St. Pierre, Adding machines and wild attractors, Ergod. Th. and Dyn. Sys. 17 (1997) 1267-1287, [.pdf]
• H. Bruin, Topological conditions for the existence of Cantor attractors, Trans. Amer. Math. Soc. 350 (1998) 2229-2263, [.pdf]
• H. Bruin, J. Hawkins, Exactness and maximal automorphic factors of unimodal maps, Ergod. Th. and Dyn. Sys. 21 (2001) 1009-1034, [.pdf]
• H. Bruin, W. Shen, S. van Strien, Existence of unique SRB-measures is typical for unimodal families, Annales Scientifiques de l'ENS, 4e serie, 39 (2006) 381-414, [.pdf]
• H. Bruin, V. Jiménez-López, On the Lebesgue measure of Li-Yorke pairs for interval maps Preprint 2009, [.pdf] Commun. Math. Phys. 299 (2010), 523--560. The final publication is available at http://www.springerlink.com/content/fu01lpu5u4305572/. (DOI: 10.1007/s00220-010-1085-9)

### Symbolic dynamics

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.

• H. Bruin, Combinatorics of the kneading map, Int. Jour. of Bifur. and Chaos 5 (1995) 1339-1349, [.pdf]
• H. Bruin, Dimensions of recurrence times and minimal subshifts, Conference Proceedings: from Chrystal to Chaos, Luminy (2000), [.pdf]
• H. Bruin, Minimal Cantor systems and unimodal maps, J. Difference Eq. and Appl. 9 (2003) 305--318, [.pdf]
• H. Bruin, O. Volkova, The complexity of Fibonacci-like kneading sequences, Theo. Comp. Science. 377 (2005) 379-389, [.pdf]

### Topological dynamics of interval maps

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.

• H. Bruin, G. Keller, M. St. Pierre, Adding machines and wild attractors, Ergod. Th. and Dyn. Sys. 17 (1997) 1267-1287, [.pdf]
• H. Bruin, Homeomorphic restrictions of unimodal maps, Contemp. Math. 246 (1999) 47-56, [.pdf]
• H. Bruin, Minimal Cantor systems and unimodal maps, J. Difference Eq. and Appl. 9 (2003) 305--318, [.pdf]
• H. Bruin, V. Jiménez-López, On the Lebesgue measure of Li-Yorke pairs for interval maps Preprint 2009, [.pdf]
• J. Bobok, H. Bruin, The topological entropy of Banach spaces Preprint 2011 [.pdf] J. Difference Equ. Appl. 18 (2012), no. 4, 569-578.

### Topological entropy of interval maps

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.

• H. Bruin, Non-monotonicity of entropy of interval maps, Phys. Lett. A 202 (1995) 359-362. Preprint version [.pdf]
• H. Bruin, An algorithm to compute the topological entropy of a unimodal map, Proceedings contribution for the conference in memory of W. Szlenk, Barcelona (1996) Internat. J. Bifur. Chaos 9 (1999) 1881-1882, [.pdf]
• H. Bruin, S. van Strien, Monotonicity of entropy for real multimodal maps Preprint 2009, revised version of December 2013 - Journ. Amer. Math. Soc. 28 (2015), no. 1, 1-61. [.pdf]
• J. Bobok, H. Bruin, Constant slope maps and the Vere-Jones classification. Preprint 2016 [.pdf]. Entropy 18(6) , 234, 2016. DOI:10.3390/e18060234

### Inverse limit spaces

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). 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.

• H. Bruin, K. M. Brucks, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999) 219-246, [.pdf]
• H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology and its Applications 96 (1999) 191-208, [.pdf]
• H. Bruin, Inverse limit spaces of post-critically finite tent maps, Fund. Math. 165 (2000) 125-138, [.pdf]
• H. Bruin, Asymptotic arc-components of unimodal inverse limit spaces, Top. & Appl. 152 (2005) 182 - 200, [.pdf]
• H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proceedings 31 (2007) no. 1, 37-50. [.pdf]
• M. Barge, H. Bruin, S. Stimac, The Ingram Conjecture [.pdf] Geom. Topol. 16 (2012), 2481-2516.
• M. Barge, H. Bruin, L. Jones, L. Sadun, Homological Pisot substitutions and exact regularity Preprint 2010, [.pdf] Isr. J. Math. 188 2012, 281-300 (DOI) 10.1007/s11856-011-0123-4
• H. Bruin, S. Stimac, On isotopy and unimodal inverse limit spaces Preprint 2011, [.pdf], Discrete and Continuous Dynamical Systems 32 (2012), no. 4, 1245-1253.
• H. Bruin, S. Stimac, Entropy of homeomorphisms on unimodal inverse limit spaces, [.pdf] Nonlinearity 26 (2013) 991-1000.

### Complex dynamics

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.

• H. Bruin, D. Schleicher, Symbolic dynamics of quadratic polynomials, Preprint (2002) This is available on the web-pages of the Mittag-Leffler institute.
• H. Bruin, D. Schleicher, Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials. Journal Lond. Math. Soc. 78 (2008), no. 2, 502--522. [.pdf]
• H. Bruin, A. Kaffl, D. Schleicher, Existence of quadratic Hubbard trees. Fund. Math. 202 (2009) 251-279, [.pdf]
• H. Bruin, D. Schleicher, Bernoulli measure of complex admissible kneading sequences, Preprint 2012 [.pdf] Ergod. Th. Dyn. Sys. 33 2013, 821-830. DOI: 10.1017/edts.2012.134

### Piesewise isometries

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.

• H. Bruin, S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Isr. J. Math. 137 (2003) 125-148, [.pdf]
• H. Bruin, A. Lambert, G. Poggiaspalla, S. Vaienti, Numerical analysis for a discontinuous rotation of the torus, Chaos 13 (2003) 558-571. The original publication is available at [http://www.springerlink.com]
• H. Bruin, Renormalisation in a class of interval translation maps of \$d\$ branches, Dynamical Systems, an international journal 22 (2007) 11 - 24. Preprint version 2006, [.pdf]
• H. Bruin, J. Deane, Piecewise contractions are asymptotically periodic, Proc. Amer. Math. Soc. 137 (2009), 1389-1395. [.pdf]
• H. Bruin, G. Clack, Inducing and unique ergodicity of double rotations Preprint 2011 [.pdf], Discrete and Continuous Dynamical Systems 32, no. 12 2012. Click here for the electronic version of the journal.