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Ergodic Theory and Dynamical Systems Seminar

The seminars will take place in Room B3.02 (unless stated otherwise)
Organizers: Polina Vytnova and Selim Ghazouani

Term 1

  • 2 October

    Richard Sharp (Warwick)
    Title: Periodic orbit growth on covers of Anosov flows
    Abstract: Suppose we have the lift of an Anosov flow to a regular cover. By considering periodic orbits intersecting some bounded region in the cover, one may define a Gurevic entropy which is less that or equal to the topological entropy of the Anosov flow. In the special case of a geodesic flow over a compact manifold with negative sectional curvatures, we have equality if and only if the cover is amenable. This result fails for other Anosov flows but we will discuss a natural generalisation. This is joint work with Rhiannon Dougall.

  • 9 October

    Michele Triestino (Dijon)
    Title: Smoothening singular group actions on manifolds
    Abstract: Motivated by the recent results around Zimmer’s program, we study group actions on manifolds, with singular regularity (we require that every element is differentiable at all but countably many points). The groups under considerations have a fixed point property, named FW, which generalizes Kazhdan’s property (T) (in particular we can consider actions of lattices in higher rank simple Lie groups).
    The main result is that if a group G has property FW, any singular action of G on a closed manifold
    1) either has a finite orbit,
    2) or is conjugate to a differentiable action, up to changing the differentiable structure of the manifold.
    This is a joint work with Yash Lodha and Nicolas Matte Bon.

  • 16 October

    Etienne Le Masson (Universite de Cergy-Pontoise)
    Title: Quantum ergodicity and random waves in the Benjamini-Schramm limit
    Abstract: One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random fields. We will show how the notion of Benjamini-Schramm convergence of manifolds (originally defined for graphs) can be used to formulate Berry’s conjecture precisely. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a much more general framework that will lead us to consider a case where the frequencies stay bounded and the size of the manifold increases instead. In this alternative setting we will explain how the ergodicity of the Gaussian wave and the mixing of the geodesic flow can be used to prove weaker forms or consequences of the random wave conjecture.
    Joint work with Miklos Abert and Nicolas Bergeron.

  • 17 October (2:00pm, B3.02)

    Hillel Furstenberg (Jerusalem)
    Title: Affine group actions
    Abstract: When X is a compact, convex space, and a group G acts on X preserving the affine structure, we speak of an "affine action", or, representation. In analogy with linear representation theory, one would like to describe all minimal — or irreducible — affine actions. We develop the theory for Lie groups and focus on PSL(2,R), noting that every bounded harmonic function in the unit disc leads to an irreducible affine representation. It turns out surprisingly that up to equivalence, this group has a unique irreducible affine representation.

  • 23 October

    Stephen Cantrell (Warwick)
    Title: Comparing word length with displacement for actions on CAT(-1) spaces
    Abstract: Suppose a hyperbolic group G acts sufficiently nicely on a complete CAT(-1) geodesic metric space X. Fix an origin for X. Each group element g in G displaces this origin by a distance comparable to the word length of g. In this talk we discuss various ways in which we can make an averaged comparison between the word length and displacement associated to the action of G on X. We will also discuss other geometrically interesting real valued functions on hyperbolic groups, for which these comparison results apply.

  • 30 October

    Dmitry Turaev (Imperial)
    Title: On wandering domains near a homoclinic tangency
    Abstract: Given a map, we define a wandering domain as an open region such that the diameter of its images by the iterations of the map shrinks to zero but the corresponding limit set is not a periodic point. It is known that many finitely smooth two-dimensional diffeomorphisms have wandering domains while it is not known if a polynomial diffeomorphism of a plane can have one. We discuss wandering domains whose limit sets is a homoclinic tangency. We show the existence of real analytic planar diffeomorphisms with wandering domains and discuss how to find wandering domains for polynomial diffeomorphisms of the three-dimensional space.

    Gabriella Keszthelyi (Rényi Institute, Budapest) — 3:00pm, B3.02
    Title: Dynamical properties of biparametric skew tent maps
    Abstract: pdf  

  • 6 November

    Federico Rodriguez-Hertz (Penn State/Lille)
    Title: Classification of Anosov actions and first cohomology
    Abstract: In the late 60’s and early 70’s it was conjectured that Anosov diffeomorphisms have an algebraic origin, indeed that Anosov diffeomorphisms are topologically conjugated to algebraic automorphisms of infranilmanifolds. In this direction, J. Franks and A. Manning showed that if the underlying manifold is an infranilmanifold the Anosov diffeomorphism is topologically conjugated to an algebraic one. When the acting group is higher rank, strong rigidity results are expected and it is conjectured that actions of higher rank lattices with an Anosov element should be on infranilmanifolds and smoothly equivalent to algebraic. In this talk I will discuss some recent advances in this direction, proving results analogous to Franks-Manning in the setting of higher rank abelian actions and higher rank lattices on semisimple Lie groups actions. This is a joint project with A. Brown and Z. Wang.

  • 13 November

    Jose Alves (Porto/Loughborough)
    Title: Entropy formula and continuity of entropy for piecewise expanding maps
    Abstract: We consider some classes of piecewise expanding maps in finite dimensional spaces with invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures. Using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to some families of piecewise expanding maps. Joint work with Antonio Pumariño.

  • 20 November

    Ariel Rapaport (Cambridge)
    Title: Dimension of planar self-affine sets and measures.
    Abstract: A compact K\subset\mathbb{R}^{2} is called self-affine if there exist affine contractions  \varphi_{1},...,\varphi_{m}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2} such that K=\cup_{i=1}^{m}\varphi_{i}(K). In the 1980s, Kenneth Falconer introduced a value s, called the affinity dimension, which is the ''expected'' value for the dimension of K. I will discuss a recent project with Mike Hochman, building on our joint work with Balázs Bárány, in which we establish the equality \mathrm{dim} K=s under mild assumptions.

  • 21 November (2:00pm, MS.05)

    Yotam Smilansky (Einstein Institute, Jerusalem)
    Title: Multiscale substitution schemes and Kakutani sequences of partitions.
    Abstract: Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.

  • 27 November

    Björn Winckler (Imperial)
    Title: Instability of renormalization
    Abstract: In this talk I will discuss renormalization of low-dimensional dynamical systems and associated phenomena such as universality and rigidity. The first part of the talk will be an introduction to the classical results in this field. In the second part I will discuss new phenomena that appear in the article 'Instability of Renormalization' (arXiv:1609.04473). In particular, I will introduce the renormalization operator acting on Lorenz maps (these are one-dimensional maps associated with three-dimensional flows undergoing a homoclinic bifurcation). This operator turns out to have non-trivial dynamics inside topological classes of stationary type which in turns leads to the phenomena of coexistence and as dimensional discrepancy. All of these notions will be explained in the talk.

  • 4 December

    Mike Whittaker (Glasgow)
    Title: Correspondences for Smale spaces
    Abstract: Correspondences provide a notion of a generalised morphism between Smale spaces. Using correspondences we define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. In this talk I will outline our construction and introduce the key ingredients. This is joint work with Robin Deeley and Brady Killough.

Term 2

  • 8 January

    Matt Galton (Warwick)
    Title: Limit theorems and the iterated weak invariance principle for slowly mixing dynamical systems.
    Abstract: The iterated weak invariance principle (WIP) is an ingredient in showing solutions of certain ODEs converge to a solution of some SDE. In this talk, we will consider dynamical systems given by a non-invertible map undergoing some mild mixing assumption with respect to bounded observables. The iterated WIP involves approximating iterated integrals of Brownian motion by iterated integrals of some process associated to the dynamical system. I will introduce the central limit theorem, WIP and the iterated WIP, and discuss sufficient conditions for these to hold.

  • 15 January

    Igors Gorbovickis (Bremen)
    Title: Critical points of the multiplier map.
    Abstract: The multiplier of a non-parabolic periodic orbit of a map z\mapsto z^2+c_0 can be extended by means of analytic continuation to a multiple-valued algebraic function on the space of all quadratic polynomials z^2+c. Information about the location of the critical points of this function might shed light on the question of possible geometric shapes of hyperbolic components of the Mandelbrot set. We show that as the period of the periodic orbits increases to infinity, the critical points of the multiplier map equidistribute on the boundary of the Mandelbrot set.
    This is joint work with Tanya Firsova.

  • 17 January (3:00pm, MS.03; joint with Geometry and Topology seminar)
    Artem Dudko (Warsaw)
    Title: On computational complexity of Cremer Julia sets.
    Abstract: Informally speaking, a compact subset of a plane is called computable if there is an algorithm which can draw arbitrarily good approximations of this set. Computational complexity measures how long does it take to draw these approximations. It is known that for some classes of rational maps (e.g. hyperbolic, parabolic and Collet-Eckmann) the Julia sets have polynomial complexity. For others (e.g. Siegel) the Julia sets can have arbitrarily high computational complexity and even may be uncomputable. However, not much is known in the case of presence of Cremer periodic points. We show that there exist abundant Cremer quadratic polynomials with Julia sets of arbitrarily high complexity.
    The talk is based on joint work with Michael Yampolsky.
  • 22 January

    Anatoly Neishtadt (Loughborough)
    Title: On destruction of adiabatic invariance in a magnetic billiard
    Abstract: Billiard in a magnetic field is a popular model in nonlinear dynamics. In this model, motion of a charged particle in a plane region with perfectly reflected smooth boundary is considered. A magnetic field is directed perpendicular to the plane. We assume that the magnetic field is strong enough and nonuniform. In the particle motion, two modes of motion alternate: the skipping along the boundary of the billiard and the drift in the interior part of the billiard. Motion at each mode has an adiabatic invariant. Change of mode of motion results in a small jump of the adiabatic invariant. We demonstrate that the accumulation of these jumps leads to the destruction of the adiabatic invariance.
    This is joint work with A. Artemyev.

  • 24 January (3:00pm, MS.03; joint with Geometry and Topology seminar)

    Dmitry Jakobson (McGill)
    Title: On small gaps in the length spectrum.
    Abstract: We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric; we discuss one result
    This is joint work with Dima Dolgopyat.

  • 29 January

    Ale Jan Homburg (Amsterdam)
    Title: On-off intermittency
    Abstract: I'll consider skew product systems (y,x) \mapsto (g(y), f_y(x)) with low dimensional maps f_y as fiber maps, driven by shifts or by expanding circle maps g. The fiber maps f_y will have a common fixed point and I will consider the case where Lyapunov exponents in the fiber direction vanish. I will explain that this can lead to intermittent dynamics.

  • 30 January (2:00pm, MS.04)

    Victor Kleptsyn (Rennes)
    Title: Interacting Polya urns
    Abstract: The classical Polya urn process is a reinforcement process, in which there are balls of different color in the urn, we take out a ball at random, and the color that was just out of it gets an advantage for all future turns: we return this ball to the urn and add another one of the same color.
    However, in this process on every step all the colors are competing. What will happen if on different steps there will be different subsets of competing colors? For instance, if there are companies that compete on different markets, or if a signal is choosing its way to travel?
    Some questions here have nice and simple answers; my talk will be devoted to the results of our joint project with Mark Holmes and Christian Hirsch on the topic.

  • 5 February (2:00pm, B3.02)

    John Smillie (Warwick)
    Title: Counting closed geodesics in polygonal billiard tables
    Abstract: I will describe how ideas involving renormalisation can be used to count the number of families of closed geodesics of a given length in planar billiard tables where the angles are rational multiples of \pi. If time permits I will explain the conjectural connection with Ratner theory.

  • 12 February

    Yuri Lima (UFC)
    Title: Markov partitions for billiards
    Symbolic dynamics is a tool that simplifies the study of dynamical systems in various aspects. One approach to obtain symbolic dynamics is constructing Markov partitions: if properly constructed, then the system has properties similar to Markov chains in probability. In this talk, we will discuss the recent developments of this approach for non-uniformly hyperbolic planar billiards. A consequence is a new estimate on the number of periodic trajectories for the billiard map of some Sinai billiards.

  • 14 February (1:00pm, MS.05)
    Italo Cipriano (Santiago)
    Title: On the Wasserstein distance between stationary probability measures
    Abstract: I will present the answers to some of the questions proposed by Jon Fraser in [First and second moments for self-similar couplings and Wasserstein distances, Mathematische Nachrichten, 288, (2015), 2028—2041]. This is in part joint work with Mark Pollicott.
  • 19 February

    Leticia Pardo Simón (Liverpool)
    Title: Escaping dynamics of a class of transcendental functions
    Abstract: As a partial answer to Eremenko's conjecture, it is known for functions with bounded singular set and of finite order that every point in their escaping set can be connected to infinity by an escaping curve. Even if those curves, called "hairs" or "rays" not always land, this has been positively proved for some functions with bounded postsingular set by showing that their Julia set is structured as a Cantor Bouquet. In this talk I will consider certain functions with bounded singular set but unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, some hairs will split when they hit critical points. We show that the existence of a map on their parameter space whose Julia set is a Cantor Bouquet guarantees that such hairs, if maybe now with split ends, still land.


    Tony Samuel (3:00pm, B1.01) (Birmingham)

    Cancelled


    Title: On the regularity of Sturmian words and other aperiodic sequences
    Abstract: The theory of aperiodic order is a relatively young field of mathematics, which has attracted considerable attention in recent years. It has grown rapidly over the past three decades; on the one hand, due to the discovery of quasicrystals; and on the other hand, due to intrinsic mathematical interest in describing the very border between crystallinity and aperiodicity. While there is currently no axiomatic framework for aperiodic order, various types of order conditions have been and are still being investigated.

    At the turn of the century Durand and Lagarias & Pleasants established key order conditions to be studied. In this talk, we will discuss these order conditions, as well as generalisations and extensions thereof, for two classes of aperiodic sequences: Sturmian words and a new family of aperiodic sequences stemming from Grigorchuk's infinite 2-group. We will also show that (exact) Jarník sets naturally give rise to a classification of Sturmian words in terms of such order conditions.

  • 26 February

    Valery Gaiko (National Academy of Sciences of Belarus, Minsk)
    Title: Limit cycles of planar polynomial dynamical systems
    Abstract: We discuss new bifurcational geometric methods based on the Wintner-Perko termination principle for the global qualitative analysis of planar polynomial dynamical systems. This is related to the solution of Hilbert’s sixteenth problem on the maximum number and distribution of limit cycles.

  • 5 March

    Vassili Gelfreich (Warwick)
    Title: Energy transfer in slow-fast systems: ergodicity vs non-ergodicity
    Abstract: In this talk we discuss the role of ergodicity in the equilibration in slow-fast systems. In classical statistical mechanics the justification of the low of equipartition relies on ergodicity of dynamics. Second, in a slow-fast system ergodicity of the fast sub-system impedes the equilibration of the whole system due to the presence of adiabatic invariants. We show that the violation of ergodicity in the fast dynamics effectively drives the whole system to equilibrium. We use some models based on non-automous billiard to illustrate the ideas.

    This talk is based on joint works with V. Rom-Kedar, K. Shah and D. Turaev.

  • 12 March

    Jens Marklof (Bristol)
    Title: Pair correlation and equidistribution on manifolds
    Abstract: A series of recent papers [1, 2, 4] has established that if a given deterministic sequence on a circle has a Poisson pair correlation measure, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori [3, 5]. The purpose of this lecture is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to sequences (and more general triangular arrays) in bounded Euclidean domains as well as compact Riemannian manifolds.
    References

    1. C. Aistleitner, T. Lachmann, and F. Pausinger, Pair correlations and equidistribution, Journal of Number Theory 182 (2018), 206–220.
    2. S. Grepstad and G. Larcher, On pair correlation and discrepancy, Arch. Math. 109 (2017), 143–149
    3. A. Hinrichs, L. Kaltenböck, G. Larcher, W. Stockinger and M. Ullrich, On a multi-dimensional Poissonian pair correlation concept and uniform distribution, arXiv:1809.05672
    4. S. Steinerberger, Poissonian pair correlation and discrepancy, Indagationes Math. 29 (2018), 1167– 1178
    5. S. Steinerberger, Poissonian pair correlation in higher dimensions, arXiv:1812.10458

Term 3

  • 30 April

    Pierre Berger (Paris)
    Title: Emergence of non-ergodic dynamics
    Abstract: Recently wild dynamics were shown locally typical of in the sense of Kolmogorov. These dynamics are wild in the sense that they seem difficult to approximate by a finitely ergodic system. In order to quantify this complexity, I introduced the emergence \mathcal E(\epsilon) as the minimal number N of probability measures (\mu_i)_{1\le i\le N} such that the empirical function x\mapsto \mathsf e_k(x):= \frac1k \sum_{i=1}^k \delta_{f^i(x)} satisfies:  \limsup_{k\to \infty} \int_M d_{W_1} (\mathsf e_k(x), \{\mu_i: {1\le i\le N}\})d Leb \le \epsilon\; where d_{W_1} is the 1-Wasserstein metric on the space of probability measures.

    I will present a program which aims to

    1. Show the typicality of dynamics with high emergence.
    2. Describe dynamics with high emergence using a dictionary with the notion of entropy.

    Such a program will be illustrated by recent achievements in several collaborations, with Bochi, Biebler, Talebi, Turaev, for symplectic, differentiable or holomorphic dynamics.

    Oliver Butterley (Nova Gorica) MS.05, 3pm
    Title: Parabolic Flows Renormalized by Partially Hyperbolic Maps
    Abstract: We will discuss 3-dimensional parabolic flows which are renormalized by circle extensions of Anosov diffeormorphisms (this includes nilflows on the Heisenberg nilmanifold). We use the spectral information of the transfer operators associated to these partially hyperbolic maps to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.

  • 7 May
    James Robinson (Wawick)
    Title: Finite-dimensional dynamics on finite-dimensional attractors
    Abstract: I will talk about reproducing the dynamics on finite-dimensional attractors for systems on infinite-dimensional spaces using dynamical systems on finite-dimensional spaces. I will show that this can be done for the attractors of homeomorphisms (although there are still some interesting open questions), and give some indications of the difficulties in trying to do the same in continuous systems.
    This is based on joint work with Jaime Sanchez-Gabites (Universidad Autonoma de Madrid)
  • 14 May
    Kenneth Falconer
    (St Andrews)
    Title: Hausdorff, box-counting and intermediate dimensions
    Abstract: Firstly we will discuss an approach to box-counting dimensions based on capacities which leads easily to projection properties and other geometric results. We will then discuss recent work with Fraser and Kempton on a continuum of ‘intermediate’ dimensions which has Hausdorff dimension at one end of the range and box-counting dimension at the other.

  • 21 May
    Toby Hall (Liverpool)
    Title: Natural extensions of unimodal maps are virtually sphere homeomorphisms
    Abstract: Let f be a unimodal map of the interval. The inverse limit construction replaces f with a self-homeomorphism - the natural extension of f — of its inverse limit space. These inverse limit spaces have complicated and intricate topology: for example, they typically contain indecomposable continua.
    I'll discuss recent work with Philip Boyland (University of Florida) and Andre' de Carvalho (University of Sao Paulo), in which we construct mild semi-conjugacies from these natural extensions to sphere homeomorphisms. The family of sphere homeomorphisms constructed in this way includes examples of Thurston's pseudo-Anosov maps; of generalized pseudo-Anosovs; and of (further generalized) measurable pseudo-Anosovs.
    Asaki Saito (Future University Hakodate) MS.05, 3pm
    Title: Pseudorandom number generation using the Bernoulli map on cubic algebraic integers
    Abstract: Pseudorandom sequences with high (dimensional) uniformity and quasirandom sequences are known to be very useful for Monte Carlo computation. It is, however, still an interesting question how we can generate a pseudorandom sequence in the original sense, i.e., a computer-generated sequence of numbers that appear similar to a typical sample of independently identically distributed random variables. According to ergodic theory, one of the simplest chaotic maps on the unit interval, namely the Bernoulli map x \mapsto 2 x \mbox{ mod } 1, can generate ideal random binary sequences. However, its use as a pseudorandom number generator has been difficult due to the drawbacks of conventional simulation methods, such as those using double-precision binary floating-point numbers or arbitrary-precision rational numbers. In this talk, we present a pseudorandom bit generator that exactly computes chaotic true orbits of the Bernoulli map on real cubic algebraic integers having complex conjugates. In particular, we clarify a seed selection method that can select initial points (i.e., seeds) without bias and can make the pseudorandom sequences derived from them be very different from each other. Moreover, in order to evaluate the memory usage of our generator, we give upper bounds concerning the growth of the representation of points on a true orbit. We also report results of a large variety of tests indicating that the generated pseudorandom sequences have good statistical properties as well as an advantage over what is probably the most popular generator, the Mersenne Twister MT19937.
    This is joint work with Akihiro Yamaguchi.
  • 28 May
    Norbert Peyerimhoff (Durham)
    Title: Minimizing length of billiard trajectories in hyperbolic polygons
    Abstract: In this talk we will play billiards inside ideal polygons of the hyperbolic plane. Together with Karl Friedrich Siburg, we conjectured a particular minimality property of closed billiard trajectories for regular ideal hyperbolic polygons and verified this conjecture in specific cases. Later, in collaboration with John Parker, we developed together a proof of this conjecture using fundamental results by Kerckhoff and Wolpert on the geodesic length function in Teichmueller space.
  • 4 June
    Jeroen Lamb (Imperial)
    Title: Classification of random circle homeomorphisms up to topological conjugacy
    Abstract: We provide a classification of random orientation-preserving homeomorphisms of the circle, up to topological conjugacy of the random dynamical systems generated by i.i.d. iterates. This classification concerns random circle homeomorphisms for which the noise space is a connected Polish space and some other minor conditions are satisfied.

    This is joint work with Thai Son Doan (Vietnam Academy of Sciences), Julian Newman (Lancaster University) and Martin Rasmussen (Imperial College London).

    Tony Samuel (Birmingham) B3.03, 3pm
    Title: On the regularity of Sturmian words and other aperiodic sequences.
    Abstract: The theory of aperiodic order is a relatively young field of mathematics which has attracted considerable attention in recent years. It has grown rapidly over the past three decades; on the one hand, due to the discovery of quasicrystals; and on the other hand, due to intrinsic mathematical interest in describing the very border between crystallinity and aperiodicity. While there is currently no axiomatic framework for aperiodic order, various types of order conditions have been and are still being studied.
    At the turn of the century Durand and Lagarias & Pleasants established key order conditions to be studied. In this talk, we will discuss these order conditions, as well as generalisations and extensions thereof, for two classes of aperiodic sequences: Sturmian words and a new family of minimal aperiodic words stemming from Grigorchuk's infinite 2-group. We will also show that (exact) Jarník sets naturally give rise to a classification of Sturmian words in terms of such order conditions as well as in terms of regularity properties of a class of spectral metrics introduced by Kellendonk and Savinien.
    This is joint work with F. Dreher, M. Gröger, M. Kessböhmer, A. Mosbach and M. Steffens.
  • 11 June
    Carl Dettmann (Bristol)
    Title: How sticky is the chaos/order boundary?
    Abstract: In dynamical systems with divided phase space, the vicinity of the boundary between regular and chaotic regions is often "sticky" that is, trapping orbits from the chaotic region for long times. Here, we investigate the stickiness in the simplest mushroom billiard, which has a smooth such boundary, but surprisingly subtle behaviour. As a measure of stickiness, we investigate P(t), the probability of remaining in the mushroom cap for at least time t given uniform initial conditions in the chaotic part of the cap. The stickiness is sensitively dependent on the radius of the stem r via the Diophantine properties of \rho = \frac{2}{\pi} \arccos r. Almost all rho give rise to families of marginally unstable periodic orbits (MUPOs) where P(t) \sim \frac{C}{t} , dominating the stickiness of the boundary. After characterising the set for which rho is MUPO-free, we consider the stickiness in this case, and where rho also has continued fraction expansion with bounded partial quotients. We show that t^2 P(t) is bounded, varying infinitely often between values whose ratio is at least \frac{32}{27}. When \rho has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a \log-periodic function. In general, we expect less regular behaviour, with upper and lower exponents lying between 1 and 2. The results may shed light on the parameter dependence of boundary stickiness in annular billiards and generic area preserving maps.
  • 18 June
    Simon Baker (Warwick)
    Title: Overlapping iterated function systems from the perspective of Metric Number Theory
    Abstract: Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the divergence of naturally occurring volume sums. Importantly this result provides a quantitative description of how the rationals are distributed within the reals. In this talk I will discuss some recent work where I prove that a similar Khintchine like phenomenon occurs typically within many families of overlapping iterated function systems. Families of iterated function systems these results apply to include those arising from Bernoulli convolutions, the {0,1,3} problem, and affine contractions with varying translation parameter.
    Time permitting I will discuss a particular family of iterated function systems for which we can be more precise. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.
  • 25 June No seminar — One Day ETDS meeting in Birmingham

Aerial photograph of Maths Houses

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