# Abstracts

Shigeki AKIYAMA akiyama@math.sc.niigata-u.ac.jp (joint with Edmund Harris)

Title: Why do piecewise isometries have Pisot scalings?

Abstract: We start with a simple system, a piecewise isometry arising from a 5-fold (and thus quadratic) discretised rotation. This system turns out to be self-inducing and metrically isometric to the 2-odometer, and a Pisot scaling constant plays an essential role. In other self-inducing systems Pisot scalings occur again. Even systems with cubic rotations, but we did not find any scalings that were not Pisot. We do not know the essential reasons why Pisot numbers appear in Piecewise isometries. This leaves many interesting open questions.

Peter ASHWIN P.Ashwin@ex.ac.uk

Title: Piecewise isometries - from the line to the plane

Abstract: Piecewise isometries in the line (interval exchanges) are dynamically well understood, but it seems to be hard to lift results or techniques to higher dimensional systems. I will discuss some attempts to do this - focussing on two classes of piecewise isometric map on the plane that can be thought of as extensions of an interval exchange transformation. These are either extensions from the circle at infinity (a cone exchange) or from a line in the plane (interval exchange with a twist). We focus on two questions - boundedness of orbits (for cone exchanges) and existence of non-smooth invariant curves (for interval exchange with a twist). This is a joint work with Arek Goetz (SFSU).

Chris Bose (TBC), Math and Stats, University of Victoria, CANADA cbose@uvic.ca

Title: Intermittent Baker's Maps

Abstract: We present a family of nonuniformly hyperbolic generalized baker's maps, deriving sharp polynomial rates for decay of correlation on 2D Holder observables. These maps arise as invertible covers for a family of nonuniformly expanding circle maps recently treated by Cristadoro, Haydn, Marie and Vaienti (2010) that were in turn based on an earlier example of Alves and Araujo (2004). The notion of first hyperbolic times can also be investigated in this context. This is joint work with Rua Murray.

Robert Fokkink R.J.Fokkink@tudelft.nl (joint with Cor Kraaikamp C.Kraaikamp@tudelft.nl)

Title: Dynamics of subtractive algorithms

Fritz Schweiger introduced the notion of a subtractive algorithm, a family of maps that generalize the euclidean algorithm.

Subtractive algorithms have interesting dynamics and exotic invariant sets. A full understanding of their dynamics appears to be difficult.

In this talk I will give an answer to some questions of Schweiger. This is joint work with Cor Kraaikamp and Hitoshi Nakada.

Arek GOETZ goetz@sfsu.edu

Title: Breaking of symmetries and open problems in two dimensional piecewise isometries.

Abstact. Piecewise isometries are dynamical systems generated by a function *T*:*X* →*X* acting as an isometry of disjoint subsets of *X*. While piecewise isometries are natural generalizations of interval exchange transformations, the landscape of phenomena is quite different. In this talk we illustrate some of the differences and we describe a few open problems. We also illustrate how symbolic computations give an insight into symmetries and their lack of in first return maps.

Shin KIRIKI (Kyoto University of Edu.)

Title: Renormalized dynamics near heterodimensional tangencies and existence of robust cycles

Abstract: We consider 3-dimensional diffeomorphisms which have a heterodimensional cycle containing a heterodimensional tangency associated with saddle periodic points. We first present renormalizations arbitrarily close to the tangency whose return maps converge to Hénon-like diffeomorphisms of index 2 with blender-horseshoes. Moreover, on such a situation, we find robust heterodimensional cycles associated with the blender-horseshoes and the continuation of one of the saddle periodic points. This is the joint work with L.J. Díaz and K. Shinohara.

Tomasz NOWICKI tnowicki@us.ibm.com

Title: Dynamics of the Error Diffusion Algorithm

1. Motivation: printing, job scheduling, chairman assignment problem. 0-1 polytopes (cubicles).
Geometry: Voronoi regions, invariant regions,planar polytopes (R^{2})
Boundedness Theorem (without proof, but with hand waving).

2. Properties of invariant regions, Inclusion on the polytope and the Voronoi corners (proofs). Topological regularity, Finite control.

3. The Case of the Acute Simplex. Tiling, subtiling and mutlitiling (proofs).

Abstract: Error Diffusion Algorithm is a greedy, online Analog-to-Discreet procedure to convert a stream of inputs coming from a (convex, compact) polytope in R^{d} into a stream of outputs from the vertices of this polytope, in such a way that the cumulative error (sum of inputs minus sum of outputs) is minimized.

The properties of the algorithm can be investigating by studying the dynamics of a map defined by piecewise translations.

Used notions: izometries, convexity, basic topology and linear algebra in vector and affine spaces.

Luciano PRUDENTE with Alexander Arbieto

Title: Existence and uniqueness of equilibrium states for some horseshoe.

Abstract: In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.

Evelyn SANDER esandergmu@gmail.com

Title: Connecting period-doubling cascades to chaos

Abstract: The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in dynamical systems varying with a parameter. Bifurcation diagrams often reveal the intermingling of cascades and chaos. Our recent research rigorously links cascades and chaos in generic families of maps in any finite dimension using a one manifold of periodic orbits in phase cross parameter space. Our examples include iterated maps arising from Poincaré sections of both finite-dimensional flows and infinite-dimensional delay-differential equations.

Duncan SANDS duncan.sands@math.u-psud.fr (only talk Tuesday / Wedneday)

Title: Dynamics of continuous piecewise affine maps

Abstract: This mini course will cover the following topics, mostly in the context of piecewise affine homeomorphisms of the plane, sometimes only for Lozi maps: piecewise affine maps; Lozi maps; compactification; piecewise projective maps; topological entropy: estimation from above - estimation from below - measures of maximal entropy - parameter dependence - loci of maximal/minimal entropy; symbolic dynamics; topological attractors; global dynamics; physical measures; Pesin theory; SBR measures.

Daniel SCHNELLMANN daniel.schnellmann@ens.fr

Title: Typical points for one-parameter families of non-degenerate piecewise expanding unimodal maps

Abstract: A piecewise expanding unimodal map on the unit interval admits a unique absolutely continuous invariant measure. By Birkhoff's Ergodic Theorem Lebesgue almost every point in the unit interval is typical for this measure, i.e., for each continuous function its time average along the forward orbit of a typical point is equal to its space average over the absolutely continuous invariant measure. Given a one-parameter family of piecewise expanding unimodal maps, we show that under a weak non-degeneracy condition the turning point is typical for the absolutely continuous invariant measure for Lebesgue almost every map in the family. This almost sure typicality result in the parameter space also applies to points different from the turning point.

Maciej P. WOJTKOWSKI wojtkowski@matman.uwm.edu.pl

Title: Symmetries of Berg partitions

Abstract: 2-element Markov partitions for 2-dimensional toral automorphisms can have additional symmetries. We give a complete classification of such symmetries (based on a joint work with Artur Siemaszko).

Christian WOLF cwolftm23@gmail.com

Title: Variation of topological pressure for complex Hénon maps

Abstract: Complex Hénon maps are the simpliest invertible complex dynamical systems with non-trivial dynamics.

In this talk we discuss how one can use results from one-dimensional complex dynamics to study the variation of the topological pressure for these maps. As a consequence we obtain the uniqueness of measures of maximal dimension and the discontinuity of the Hausdorff dimension of Julia sets at the boundary of the hyperbolicity locus.

Izzet Burak YILDIZ yildiziz@math.msu.edu

Title: Topological entropy of the Lozi maps

Abstract: We introduce some recent results about the topological entropy of the Lozi maps which are piecewise-affine analogues of the Henon maps. In particular, we describe the monotonicity and discontinuity properties of the entropy. For the monotonicity results, we make use of the symbolic dynamics and the pruning theory. For the discontinuity results, we describe how the entropy can jump up at certain parameters.

Yiwei Zhang (joint with Congping Lin) yz297@exeter.ac.uk

Title: Invariant measures with bounded variation densities for piecewise area preserving maps

Abstract: We investigate the properties of absolutely continuous invariant probability measures (ACIPs) for piecewise area preserving maps (PAPs) on $\mathbb{R}^d$. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. In particular for PWIs, we use functional approach to explore the relationship between topological transitivity and uniqueness of ACIPs, especially those measures with bounded variation densities. This "partially'' answers one of the fundamental questions posed in {Goetz03} - determine all invariant non-atomic probability Borel measures in piecewise rotation. When reducing to interval exchange transformations (IETs), we demonstrate that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have a very irregular density (namely of unbounded variation and discontinuous everywhere) and intermingle with each other.

A. Goetz, Piecewise Isometries - An Emerging Area of Dynamical Systems, in *Fractals in Graz* (2001), ed. P. Grabner and W. Woess, *Trends Math., Birkhauser, Basel* (2003), 133-144.