Advances in Ergodic Theory, Hyperbolic Dynamics and Statistical Laws

The Australian National University

28 November – 2 December 2016

Sponsors

MATHSFEST Program28 November – 2 December 2016

Time Monday Tuesday Wednesday Thursday Friday

8:30Registration

9:00 Invited LectureF. Ledrappier

Invited LectureD. Kelly

Invited LectureC. Liverani

Invited LectureF. Pène

Invited LectureK. Ramanan

10:00Coffee Break Coffee Break Coffee Break Coffee Break Coffee Break

10:30 Invited LectureB. Goldys

Invited LectureM. Dellnitz

Invited LectureM. Pollicott

Invited LectureK. Burns

T. Dooley

11:00A. Blumenthal

11:30S. Balasuriya J. Wouters K. Yamamoto V. Climenhaga A. Sims

12:00P. Koltai C. Kalle I. Rios J. F. Alves E. G. Altmann

12:30Lunch Lunch Lunch Lunch Lunch

14:00 Invited LectureA. Hassell

Invited LectureM. J. Pacifico

Invited LectureA. Wilkinson

15:00Coffee Break Coffee Break Coffee Break

15:30D. Dragičević S. Hittmeyer T. Schindler

16:00H. Osinga S. V. da Silva M. Field

16:30M. Nicol

17:00K. Burns event

18:00Posters/Drinks/BBQ

Practical Information

VenueWelcome to the Math(s)Fest! The talks for the Advances in Ergodic Theory, Hyperbolic Dynamicsand Statistical Laws Workshop will all be held in the main hall of University House on the ANUcampus.

This booklet includes a map of campus and University House is in the top right corner of square C2.

EventsOn Monday evening from 18:00 to 22:00, the workshop is hosting a welcome barbeque and postersession at University House. If you would like to bring an extra guest to the barbeque, please let usknow as soon as possible. The cost per extra guest will be AUD $60.

On Thursday evening at 17:00, the workshop is celebrating the Keith Burns’ sixtieth birthday in theCommon Room of University House. Light snacks and, of course, cake will be served. There will alsobe a cash bar.

Free AfternoonsNo talks are scheduled for Wednesday and Friday afternoons and there are a number of attractionsnear the ANU campus. The National Museum of Australia is just south of campus and in easy walkingdistance. Further south, on the other side of the river, are the National Gallery of Australia and theOld and New Parliament Buildings. We plan to put more information up on the workshop website.

There is also a tourist bus line (Route 81/981) and a free city loop bus (Route 101).

InternetEduroam is available throughout the ANU campus. If you cannot access the internet throughEduroam, talk to one of the organisers to get access to the wi-fi at University House.

AccommodationIf the workshop has arranged your accommodation, then you should have received an email from uswith details. If you unsure of your check-in details, please talk to one of the organizers or contact usat dynamics.mathfest.australia@gmail.com.

DiningThere are a number of restaurants on campus, as noted by the fork and knife logos on the campusmap. Most of them are in and around Union Court (square C3 on the map).

East of the ANU Campus are a number of restaurants in and around the large “Canberra Center”shopping complex. This complex also has a supermarket for buying groceries. This area is about atwenty minute walk from University House.

We hope you have a great time in Canberra!

Organising CommitteeGary FroylandCecilia González-TokmanGeorg GottwaldAndy HammerlindlMatthew NicolLuchezar Stoyanov

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List of Registrants

Current as of Thu 10 Nov 2016

Assoc Prof Eduardo G. Altmann University of SydneyProf José F. Alves University of PortoMr Fadi Antown University of New South WalesAssoc Prof Sanjeeva Balasuriya The University of AdelaideMs Chantelle Louise Blachut The University of QueenslandDr Alex Blumenthal University of Maryland College ParkProf Keith Burns Northwestern UniversityMr Timothy Peter Bywaters University of SydneyProf Vaughn Climenhaga University of HoustonMr Peter Cudmore The University of QueenslandProf Michael Dellnitz University of PaderbornDr Davor Dragičević University of New South WalesDr Nazife Erkursun Ozcan Hacettepe UniversityProf Michael John Field Imperial College LondonProf Gary Froyland University of New South WalesProf Beniamin Goldys University of SydneyDr Cecilia González-Tokman The University of QueenslandProf Georg Gottwald University of SydneyMr Richard Greenhalgh The University Of Western AustraliaDr Andy Hammerlindl Monash UniversityProf Andrew Hassell Australian National UniversityDr Stefanie Hittmeyer The University of AucklandMr Kieran Jarrett University of BathDr Charlene Kalle University of LeidenDr David Kelly New York UniversityDr Robert Kenny The University Of Western AustraliaMr Alexander Khor The University Of Western AustraliaDr Péter Koltai Freie Universität BerlinDr Bernd Krauskopf University of AucklandMr Eric Kwok University of New South WalesProf François Ledrappier Université Paris 6Prof Carlangelo Liverani Università degli Studi di Roma “Tor Vergata”Mr Michael Arthur Mampusti University of WollongongDr Gemma Mason The University of AucklandMs Fakhereh Mohammad esmaeili fakhereh University of HyderabadMr Jack Murdoch Moore The University Of Western AustraliaDr Rua Murray University of CanterburyProf Matthew James Nicol University of HoustonMr Adam Nie Australian National UniversityMr Daniel Nix Australian National UniversityProf Hinke Osinga The University of AucklandDr Maria Jose Pacifico Universidade Federal do Rio de JaneiroProf Jayantha Pasdunkorale Arachchige University of RuhunaProf Françoise Pène Université de Bretagne OccidentaleProf Mark Pollicott University of WarwickProf Kavita Ramanan Brown UniversityProf Isabel Rios Universidade Federal FluminenseDr Tanja Schindler Australian National UniversityDr Simone Vasconcelos da Silva Universidade de Braśılia

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List of Registrants

Prof Aidan Sims University of WollongongProf Luchezar Stoyanov The University Of Western AustraliaDr Melissa Tacy Australian National UniversityDr Ilknur Tulunay University of Technology, SydneyDr Petrus van Heijster Queensland University of TechnologyProf Amie Wilkinson University of ChicagoMr John Wormell University of SydneyDr Jeroen Wouters University of SydneyDr Kenichiro Yamamoto Nagaoka University of TechnologyDr Yiwei Zhang Huazhong Univ of science and technology

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1. Talks

1.1. Sampling Rare Trajectories in Chaotic System

Eduardo G. Altmann (University of Sydney)

12:00 Fri 2 December 2016 –

Assoc Prof Eduardo G. Altmann

I will present a general framework to search andsample rare trajectories in chaotic systems. Themain result is to show how properties of the sys-tem (e.g., Lyapunov exponents, fractality) shouldbe included in the proposal step of Markov ChainMonte Carlo methods. Results are shown for twoproblems: (i) computing long-lived trajectories intransient-chaos problems; and (ii) sampling tra-jectories with atypical chaoticity, as measured byan atypically large or small finite-time Lyapunovexponent. We confirm numerically for differentclasses of (high-dimensional) chaotic systems thatour algorithm scales polynomially with the ob-servable, a dramatic improvement over the ex-ponential scaling obtained in traditional uniform-sampling methods.

1.2. Gibbs-Markov-Young structures for partiallyhyperbolic attractors

José F. Alves (University of Porto)

12:00 Thu 1 December 2016 –

Prof José F. Alves

Gibbs-Markov-Young structures have been fre-quently used in last decades as an important toolfor studying the statistical properties of systemswith some hyperbolicity. In articular, the decayof correlations and large deviations can be deter-mined by the tail of recurrence times associatedto these structures. In the case of partially hy-perbolic attractors, some results have appearedin the literature, depicting the whole scenario ina fairly reasonable way in the mostly expandingcase. The mostly contracting case is still far frombeing completely understood. In this talk we willpresent some recent results in both directions.

1.3. Hyperbolic neighbourhoods in nonautonomousflows

Sanjeeva Balasuriya (The University of Adelaide)

11:30 Mon 28 November 2016 –

Sanjeeva Balasuriya, Rahul Kalampattel and

Nicholas Ouellette

Hyperbolic trajectories in nonautonomous flowshave the property that nearby trajectories expe-rience exponential separation in the relevant timedirection. Motivated by the stretching of fluid el-ements in fluid flows, here we analyse the zoneof influence of hyperbolic trajectories. We definethis to be a Hyperbolic Neighbourhood (HN), atime-varying region anchored to each time-varying

hyperbolic trajectory. We establish, using boththeoretical arguments and empirical verificationfrom model and experimental data, that the HNsprofoundly impact the Lagrangian stretching ex-perienced by fluid elements. In particular, weshow that fluid elements traversing a flow experi-ence exponential boosts in stretching while withinthese time-varying regions, that greater residencetime within HNs is directly correlated to largerFinite-Time Lyapunov Exponent (FTLE) values,and that FTLE diagnostics are reliable only whenthe HNs have a geometrical structure that is reg-ular in a specific sense.

1.4. Lyapunov exponents for small randomperturbations of predominantly hyperbolictwo-dimensional volume-preservingdiffeomorphisms, including the Standard Map

Alex Blumenthal (University of Maryland CollegePark)

11:00 Fri 2 December 2016 –

Dr Alex Blumenthal

An outstanding problem in smooth ergodic the-ory is the estimation from below of Lyapunov ex-ponents for maps which exhibit hyperbolicity ona large but non-invariant subset of phase space.It is notoriously difficult to show that Lypaunovexponents actually reflect the predominant hyper-bolicity in the system, due to cancellations causedby the ’switching’ of stable and unstable directionsin those parts of phase space where hyperbolicityis violated.

In this talk I will discuss the inherent difficulties ofthe above problem, and will discuss recent resultswhen small IID random perturbations are intro-duced at every time-step. In this case, we are ableto show with relative ease that for a large class ofvolume-preserving predominantly hyperbolic sys-tems in two dimensions, the top Lypaunov expo-nent actually reflects the predominant hyperbol-icity in the system. Our results extend to thewell-studied Chirikov Standard Map at large cou-pling. This work is joint with Lai-Sang Young andJinxin Xue.

1.5. The Weil Petersson geodesic flow

Keith Burns (Northwestern University)

10:30 Thu 1 December 2016 –

Prof Keith Burns

I will talk about the geodesic flow for the Weil-Petersson metric on the moduli space of a sur-face that supports hyperbolic metrics. This is aRiemannian metric with negative sectional curva-tures. However the classical results of Anosov donot apply because the metric is incomplete and

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1. Talks

the sectional curvatures and their derivatives arenot uniformly bounded. It was not until the 21stcentury that this geodesic flow was shown to bemixing (and in fact Bernoulli). I will give someideas from the proof and also from more recentwork directed towards showing that the flow isexponentially mixing in case of the torus with onepuncture. This is joint work with Howard Masur,Carlos Matheus, and Amie Wilkinson.

1.6. Specification and Markov properties in shiftspaces

Vaughn Climenhaga (University of Houston)

11:30 Thu 1 December 2016 –

Asst Prof Vaughn Climenhaga

Existence, uniqueness, and statistical propertiesof invariant measures for dynamical systems withhyperbolic behaviour can be studied by construct-ing a certain “tower” that represents the systemas a suspension over the full shift on a countablealphabet. I will discuss recent results that permitthis construction to be carried out under relativelyweak hypotheses.

1.7. Glimpse of the Infinite - the Approximation ofInvariant Sets for Delay and Partial DifferentialEquations

Michael Dellnitz (University of Paderborn)

10:30 Tue 29 November 2016 –

Prof Michael Dellnitz

In this talk we present a novel numerical frame-work for the computation of finite dimensionalinvariant sets for infinite dimensional dynamicalsystems. With this framework we extend classicalset oriented numerical schemes (for the computa-tion of such objects in finite dimensions) to theinfinite dimensional context. The underlying ideais to utilize appropriate embedding techniques forthe reconstruction of invariant sets in a certain fi-nite dimensional space. Finally, we illustrate ourapproach by the computation of attractors bothfor delay and for partial differential equations.

1.8. Random Group Actions

Anthony Dooley (University of Technology Sydney)

10:30 Fri 2 December 2016 –

Prof Anthony Dooley

A random group action is a generalisation ofArnold’s theory of random dynamical systems,where the time dynamics is replaced by a groupaction G and the noise is non-singular under theaction of G.

In particular, with GuoHua Zhang, we showed,for the action of a discrete amenable group, thatrandom actions have the structure of a continuousbundle, realising the structure as a skew productgenerated by the cocycle. We also developed a

theory of local fibre topological pressure, local en-tropy and a variational principle (maximum localentropy principle).

1.9. Limit theorems for random Lasota-Yorke mapsusing the spectral method

Davor Dragičević (University of New South Wales)

15:30 Mon 28 November 2016 –

Dr Davor Dragičević

The so-called Nagaev-Guivarc’h spectral methodis one of the most powerful tools in establishinglimit theorems and has been successfully appliedin various situations including Markov chains anddynamical systems. In the standard deterministicsetting the method consists of finding an appro-priate Banach space on which the transfer opera-tor has a spectral gap and then applying variousresults from the perturbation theory of linear op-erators.

In this talk, I will discuss an ongoing researchproject which aims to extend this method to therandom case. We will also explain how the multi-plicative ergodic theorem naturally appears in thissetting. This is a joint work with G. Froyland, C.González Tokman and S. Vaienti.

1.10. Functional Networks

Mike Field (Imperial College London)

16:00 Thu 1 December 2016 –

Prof Mike Field

We describe a result that yields a functional de-composition for a large class of networks. Theresult allows us to express the function of a dy-namic network in terms of dynamics of individualnodes and constituent subnetworks.

Nonsmooth effects, such as changes in networktopology through decoupling of nodes and stop-ping and restarting of nodes, are one of the cru-cial ingredients needed for this result. In net-works modelled by smooth dynamical systems, allnodes are effectively coupled to each other at alltimes and information propagates instantly acrossthe entire network. A spatiotemporal decomposi-tion is only possible if the network dynamics isnonsmooth and (subsets of) nodes are allowed toevolve independently of each other for periods oftime. This allows the identification of dynamicalunits, each with its own function, that togethercomprise the dynamics and function of the entirenetwork. The result highlights a drawback of av-eraging over a network: the loss of informationabout the individual functional units, and theirtemporal relations, that yield network function.

We conclude with questions about how to con-struct continuum models (smooth or stochasticPDEs) for functional networks that allow us to

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regain smoothness in a context where nonsmooth-ness is an inherent part of the network function.

1.11. Some ergodic problems for stochastic PDEs

Beniamin Goldys (University of Sydney)

10:30 Mon 28 November 2016 –

Prof Beniamin Goldys

I will present some results on long time be-haviour of systems evolving randomly in space andtime described as solutions to stochastic evolutionequations on.

1.12. Dynamical systems theory applied toeigenfunction estimates

Andrew Hassell (Australian National University)

14:00 Mon 28 November 2016 –

Prof Andrew Hassell

In this talk, I will survey how results from dynam-ical systems have been used to prove estimates forhigh-energy eigenfunctions on Riemannian mani-folds.

1.13. The geometry of blenders in athree-dimensional Hénon-like family

Stefanie Hittmeyer (The University of Auckland)

15:30 Tue 29 November 2016 –

Dr Stefanie Hittmeyer

Blenders are a geometric tool to construct com-plicated dynamics in a theoretical setting. Weconsider an explicit family of three-dimensionalHénon-like maps that exhibit blenders in a spe-cific regime in parameter space. Using advancednumerical techniques we compute stable and un-stable manifolds in this system and present oneof the first numerical pictures of the geometry ofblenders.

1.14. Matching for generalisedbeta-transformations

Charlene Kalle (University of Leiden)

12:00 Tue 29 November 2016 –

Dr Charlene Kalle

The phenomenon of matching (i.e., merging ofleft/right limits at a discontinuity after some it-erates) has been observed in many families with,in some sense, nice algebraic properties. It turnout that also in the family x 7→ βx + a mod 1,matching occurs for many values of β. In thistalk I would like to describe the mechanisms be-hind this. This is based on joint work with HenkBruin and Carlo Carminati.

1.15. Ergodic theory in data assimilation

David Kelly (New York University)

9:00 Tue 29 November 2016 –

Dr David Kelly

Data assimilation describes the method of blend-ing dynamical models and observational data,with the objective of reducing uncertainty in stateestimation and prediction. The procedure hasan ‘optimal’ Bayesian solution, which tends to becomputationally intractable for high dimensionalmodels. As a consequence, many approxima-tion procedures, called approximate filters, havebeen developed in the geoscience and numericalweather prediction communities, where modelstend to be very high dimensional, and where stateestimation and uncertainty quantification are cen-tral tenets. It is important that we judge theseapproximations on how well they inherit impor-tant features from the true Bayesian solution. Inthis talk, we will investigate ergodicity for sev-eral types of filters that are prevalent in numericalweather prediction. Ergodicity is of crucial impor-tance for filters; it implies a robustness with re-spect to initial perturbations in state approxima-tions, moreover it suggests that the filter, whichis a proxy for the true underlying dynamical sys-tem, is inheriting important physical statisticalproperties. Alongside mostly positive results, wewill see that approximate filters don’t always doa good job of inheriting ergodicity. We define aclass of models, which are highly stable (and cer-tainly ergodic) for which well trusted approximatefilters exhibit strong sensitivity to initialization.In other words, the filters quickly lose touch withreality. This talk is based on several joint workswith Andy Majda, Andrew Stuart, Xin Tong, EricVanden-Eijnden and Jonathan Weare.

1.16. Spatio-temporal computational methods forcoherent sets

Péter Koltai (Freie Universität Berlin)

12:00 Mon 28 November 2016 –

Dr Péter Koltai

The decomposition of the state space of a dy-namical system into metastable sets is impor-tant for understanding its essential macroscopicbehavior. The concept is quite well understoodfor autonomous dynamical systems, and recentlygeneralizations appeared for non-autonomous sys-tems: coherent sets. Aiming at a unified the-ory, in this talk we first present connectionsbetween the measure-theoretic autonomous andnon-autonomous concepts. We will do this byconsidering the augmented state space, and ourresults will be restricted to systems with time-periodic forcing. Second, we introduce a data-based method to compute coherent sets. Ituses diffusion maps for spatio-temporal trajectory

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1. Talks

data, and it is consistent in the infinite-data limitwith the widely-used transfer operator approachfor coherent sets.

1.17. Local Limit Theorem in negative curvature

François Ledrappier (Université Paris 6)

9:00 Mon 28 November 2016 –

Prof François Ledrappier

We consider the universal cover of a compact man-ifold with negative curvature. We give a preciseequivalent of the heat kernel p(t, x, y) as t goes toinfinity. This is a joint work with Seonhee Lim(Seoul Nat. Univ.).

1.18. Hyperbolic billiards

Carlangelo Liverani (U. Roma “Tor Vergata”)

9:00 Wed 30 November 2016 –

Prof Carlangelo Liverani

The rigorous study of hyperbolic billiards wasstarted by Sinai in 1970 and since then it hasgrown in many directions due to the importance ofthis seemingly simple model for many branches ofmathematics and physics. Today it is still a veryactive field of research. In fact, in the last yearsseveral new results have been obtained and newapplications have been proposed. I will present anidiosyncratic selection of results and open prob-lems concerning hyperbolic billiards.

1.19. Intrinsic excitability and the role of saddleslow manifolds

Hinke Osinga (University of Auckland)

16:00 Mon 28 November 2016 –

Prof Hinke Osinga

Excitable cells, such as neurons, exhibit complexoscillations in response to external input, e.g.,from other neurons in a network. We considerthe effect of a brief stimulation from the rest stateof a minimal neuronal model with multiple timescales. The transient dynamics arising from suchshort current injections brings out the intrinsicbursting capabilities of the system. We focus ontransient bursts, that is, the transient generationof one or more spikes, and use a simple polynomialmodel to illustrate our analysis. We take a geo-metric approach to explain how spikes arise andhow their number changes as parameters are var-ied. We discuss how the onset of new spikes iscontrolled by stable manifolds of a slow manifoldof saddle type. We give a precise definition of sucha stable manifold and use numerical continuationof suitable two-point boundary value problems toapproximate them.

1.20. Lagrange and Markov dynamical spectra ofLorenz-like attractors

Maria José Pacifico (Universidade Federal do Riode Janeiro)

14:00 Tue 29 November 2016 –

Dr Maria José Pacifico

In a joint work with S. Romaño and C. G. Moreira,we prove that the Hausdorff dimension of a geo-metric Lorenz attractor is strictaly greater than2. From this, we conclude that the interior of theLagrange dynamical spectra as well the interiorof the Markov dynamical spectra of a geometricLorenz attractor is non empty.

1.21. Quantitative recurrence for slowly mixinghyperbolic systems

Françoise Pène (Université de BretagneOccidentale)

9:00 Thu 1 December 2016 –

Prof Françoise Pène

We are interested in the counting process of visitsto a small around the origin, and more preciselyin its behaviour as the radius of the ball goes to 0.We prove the convergence in distribution of thisprocess (under a suitable time normalization) tothe standard Poisson process. This is a joint workwith Benôıt Saussol.

1.22. Rigorous estimates for the Lanford map

Mark Pollicott (University of Warwick)

10:30 Wed 30 November 2016 –

Prof Mark Pollicott

The Lanford map is a simple example of a map ofthe interval given by

Tx = 2x+ 0.5x(1− x) (mod 1).We use this as a test case to illustrate how cer-tain natural characteristics (e.g., variance, linearresponse) can be rigorously estimated (with errorestimates) using ideas from thermodynamic for-malism. This is joint work with Oliver Jenkinsonand Polina Vytnova.

1.23. Decay of Correlations in various Hard-Coremodels

Kavita Ramanan (Brown University)

9:00 Fri 2 December 2016 –

Prof Kavita Ramanan

Recently a strong connection has been establishedfor certain discrete models between the algorith-mic problem of counting on graphs and a certaincorrelation decay property. In particular, thishas been established for the so-called hard-coremodel that arises in statistical physics, telecom-munications and combinatorics, and describes a

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1. Talks

one-parameter family of probability measures, in-dexed by a positive activity parameter, on 0-1 con-figurations on a graph. We present several resultson variants of the standard hard-core model, andshow how dynamical systems techniques are use-ful for establishing the correlation decay property.We especially focus on a continuous version ofthe model, which is motivated by the algorithmicproblem of computing the volume of a polytope.The talk is based on joint work with D. Gamarnik.

1.24. Decreasing entropy in the destruction ofhorseshoes via internal tangencies

Isabel Rios (Universidade Federal Fluminense)

12:00 Wed 30 November 2016 –

Prof Isabel Rios

We study the variation of the topological entropyfor a family of horseshes bifurcating an internaltangency. We prove that, restricted to a set ofparameters with total density at the breaking-contact bifurcation, the topological entropy is anon-increasing function. This result formalizesthe idea that the loss of transversal homoclinic in-tersections implies the decreasing of the amountof dynamics. This is a joint work with A. de Car-valho, L. J. Dı́az and K. Dı́az-Ordaz.

1.25. Generalized strong law for intermediatelytrimmed Birkhoff sums

Tanja Schindler (Australian National University)

15:30 Thu 1 December 2016 –

Tanja Schindler

By a theorem by Aaronson there is no strong lawof large numbers for observables of a dynamicalsystem with infinite expectation. However, fora suitable distribution function, the sum is es-sentially dominated by the contribution of a fi-nite number of its largest terms and a solutioncan be obtained from a generalised strong lawof large numbers for the lightly trimmed sum,that is, the Birkhoff sum

∑nk=1 ϕ ◦T k−1 trimmed

by a finite number of extremes. More formally,we consider a permutation π ∈ Sn such thatϕ ◦ Tπ(1)−1 ≥ . . . ≥ ϕ ◦ Tπ(n)−1 and consider thesum Srn :=

∑nk=r+1 ϕ ◦ Tπ(n)−1. We say that the

system fulfils a lightly trimmed strong law, if thereexist a real-valued sequence (dn) and r ∈ N suchthat limn→∞ Srn/dn = 1 almost surely.

There is a vast literature covering the case oflightly trimmed strong law for independent as wellas for mixing processes, one example in this areais the lightly trimmed strong law for the digits ofthe continued fractions transformation.

A generalisation, applicable for general distribu-tion functions of a non-negative observable, isto use intermediate trimming, that is, consider

the trimmed Birkhoff sum Sbnn :=∑nk=bn+1

ϕ ◦Tπ(k)−1 with bn tending to infinity and bn = o (n).

The proof is based on a large deviation result fortransfer operators with spectral gap.

These results can be applied to observables onsubshifts of finite type and on piecewise expand-ing interval maps. In this talk I will concentrateon the dynamical results.

This is joint work with Marc Kesseböhmer.

1.26. Diffusive - Ballistic phase transition in aRandon Polymer model

Simone Vasconcelos da Silva (Universidade deBraśılia)

16:00 Tue 29 November 2016 –

Dr Simone Vasconcelos da Silva

Phase transition issues are addressed for randompolymers on the bidimensional lattice with self-repulsive interactions. By comparison with a longrange one-dimensional ferromagnetic Ising model,it is shown that in the absence of drift and withpower law interactions, the polymer exhibits tran-sition from diffusive to a ballistic behavior. Whennon-null drifts are added and positive translationinvariant interactions are considered, the polymerpresents a ballistic behavior. We also derive aCentral Limit Theorem for the model.

1.27. Equilibrium states on noncommutativesolenoids

Aidan Sims (University of Wollongong)

11:30 Fri 2 December 2016 –

Prof Aidan Sims

KMS equilibrium states for C∗-algebraic dynami-cal systems were originally motivated by physics,but have recently re-emerged as interesting in-variants from a purely mathematical standpoint.For example, the KMS structure of the Toeplitzalgebras of irreducible directed graphs is closelytied up with Perron-Frobenius theory for the ad-jacency matrices of the graphs, and encodes datalike the Perron-Frobenius eigenvector, the spectralradius and the period of the matrix.

I’ll discuss joint work with Nathan Brownloweand Mitch Hawkins on the KMS structure of theToeplitz algebras of “noncommutative solenoids”.These are are direct limits of irrational-rotationalgebras introduced by Packer and Latrémolière,and are not accessible via standard techniquesfor KMS theory because of the interplay betweenthe subinvariance relations for the approximatingirrational-rotation algebras, and the consistencyconditions imposed by the way these subalgebrasnest. But a bit of work revealed a pretty picture:the KMS-simplex of a Toeplitz noncommutativesolenoid is isomorphic to the simplex of Borel

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probability measures on the classical solenoid. I’lltry to indicate why.

1.28. Beyond the limit of infinite time-scaleseparation: Edgeworth approximations andhomogenisation

Jeroen Wouters (University of Sydney)

11:30 Tue 29 November 2016 –

Dr Jeroen Wouters

Homogenization has been widely used in stochas-tic model reduction of slow-fast systems, includ-ing geophysical and climate systems. The the-ory relies on an infinite time scale separation. Inthis talk we present results for the realistic caseof finite time scale separation. In particular, weemploy Edgeworth expansions as finite size cor-rections to the central limit theorem and showimproved performance of the reduced stochasticmodels in numerical simulations.

1.29. Large deviations beyond specification

Kenichiro Yamamoto (Nagaoka University ofTechnology)

11:30 Wed 30 November 2016 –

Dr Kenichiro Yamamoto

In this talk, we give a criterion to determine thelarge deviation rate functions for certain dynami-cal systems without the specification property.

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2. Posters

2.1. Optimizing Linear Response for Discrete TimeHomogeneous Markov Chains

Fadi Antown (University of New South Wales)

18:00 Mon 28 November 2016 –

Mr Fadi Antown

2.2. Nonlinearly coupled oscillators.

Peter Cudmore (The University of Queensland)

18:00 Mon 28 November 2016 –

Mr Peter Cudmore

2.3. Non-singular group actions: the ergodictheorem and the critical dimension

Kieran Jarrett (University of Bath)

18:00 Mon 28 November 2016 –

Mr Kieran Jarrett

2.4. Fractal trees and the geometry of tilings

Michael Mampusti (University of Wollongong)

18:00 Mon 28 November 2016 –

Mr Michael Mampusti

2.5. TBA

Eric Kwok (University of New South Wales)

18:00 Mon 28 November 2016 –

Mr Eric Kwok

2.6. Fast and accurate approximation techniquesfor intermittent maps

John Wormell (University of Sydney)

18:00 Mon 28 November 2016 –

John Wormell

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Index of Speakers

Altmann, Eduardo G. (1.1), 4

Alves, José F. (1.2), 4Antown, Fadi (2.1), 10

Balasuriya, Sanjeeva (1.3), 4Blumenthal, Alex (1.4), 4

Burns, Keith (1.5), 4

Climenhaga, Vaughn (1.6), 5

Cudmore, Peter (2.2), 10

Dellnitz, Michael (1.7), 5Dooley, Anthony (1.8), 5

Dragičević, Davor (1.9), 5

Field, Mike (1.10), 5

Goldys, Beniamin (1.11), 6

Hassell, Andrew (1.12), 6

Hittmeyer, Stefanie (1.13), 6

Jarrett, Kieran (2.3), 10

Kalle, Charlene (1.14), 6Kelly, David (1.15), 6

Koltai, Péter (1.16), 6

Kwok, Eric (2.5), 10

Ledrappier, François (1.17), 7

Liverani, Carlangelo (1.18), 7

Mampusti, Michael (2.4), 10

Osinga, Hinke (1.19), 7

Pène, Françoise (1.21), 7

Pacifico, Maria José (1.20), 7

Pollicott, Mark (1.22), 7

Ramanan, Kavita (1.23), 7

Rios, Isabel (1.24), 8

Schindler, Tanja (1.25), 8

Silva, Simone Vasconcelos da (1.26), 8Sims, Aidan (1.27), 8

Wormell, John (2.6), 10Wouters, Jeroen (1.28), 9

Yamamoto, Kenichiro (1.29), 9

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