### Dynamics Archives for Academic Year 2012

#### No Seminar

When: Thu, September 6, 2012 - 2:00pm
Where: Math 1311

#### Noncollision singularities in a planar two-center-two-body problem

When: Thu, September 13, 2012 - 2:00pm
Where: Math 1311
Speaker: Jinxin Xue (University of Maryland)
Abstract: In this work we study a model called planar 2-center-2-body problem. In the plane, we have two fixed centers Q_1=(-\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses \mu. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painleve conjecture. This is a joint work with Dmitry Dolgopyat.

#### Typical dynamics of non-autonomous ODEs of special form

When: Thu, September 20, 2012 - 2:00pm
Where: Math 1311
Speaker: Mahesh Nerurkar (Rutgers-Camden and UMCP)
Abstract: http://www.math.umd.edu/~mmb/nerurkar.pdf

#### Path methods for strong shift equivalence of positive matrices

When: Thu, September 27, 2012 - 2:00pm
Where: Math 1311
Speaker: Mike Boyle (UMCP) - http://www.math.umd.edu/~mmb
Abstract: This is a report on joint work with Hang Kim (now deceased) and Fred Roush. In the early 1990s, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring U of R. We extend this theory. For any U, and any positive matrices A,B SSE over U and having just one nonzero eigenvalue, we show A and B are SSE over U_+. Also, positive matrices on a path of positive matrices shift equivalent over R must be SSE over R_+. A consquence is that two positive rational matrices SSE over R_+ must be SSE over Q_+. Also, for any U and n, the set of positive nxn matrices over U which are conjugate over U contains only finitely many SSE-U_+ classes.

#### No seminar

When: Thu, October 4, 2012 - 2:00pm
Where:

#### Conference at PSU

When: Thu, October 11, 2012 - 2:00pm
Where:
Speaker: NO MEETING: GO TO PSU DYNAMICS no spekaer http://www.math.psu.edu/dynsys/dw_2012/

#### TBA

When: Thu, October 18, 2012 - 2:00pm
Where: Math 1311
Speaker: Kostya Medynets (U.S. Naval Academy)
Abstract: http://www.math.umd.edu/~mmb/abstractmedynetsoct2012.pdf

#### No seminar (Colloquium)

When: Thu, October 25, 2012 - 2:00pm
Where: Math 1311

#### Lyapunov exponents of random composition of diffeomorphisms of the torus

When: Thu, November 1, 2012 - 2:00pm
Where: Math 1311
Speaker: Dominique Malicet (Universite' Pierre et Marie Curie (P6)) -
Abstract: We consider a random independent composition of a finite number diffeomorphisms of the torus (of dimension 1 or 2) close to translations, and under a diophantine condition we prove that at least one associated Lyapunov exponent is negative unless all the diffeomorphisms are simultaneously conjugated to simpler ones (translations in dimension 1 and skew products in dimension 2).

#### Oscillatory motion for the restricted planar circular three body problem

When: Thu, November 8, 2012 - 2:00pm
Where: Math 1311
Speaker: Marcel Guardia (University of Maryland) - www2.math.umd.edu/~mguardia/

Abstract: In 1980 J. Llibre and C. Simo proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume that the ratio between the masses of the two primaries was exponentially small with respect to the angular momentum (or the Jacobi constant). In the present work, we generalize their result proving the existence of oscillatory motions for any value of the mass ratio. This is a joint work with P. Martin and T. M. Seara.

#### The Statistics of Self-Intersections of Closed Curves on Orientable Surfaces

When: Thu, November 15, 2012 - 2:00pm
Where: Math 1311
Speaker: Matthew Wroten (Stony Brook)- mwroten@math.sunysb.edu

Abstract: Oriented loops on an orientable surface are, up to equivalence by free homotopy, in one-to-one correspondence with the conjugacy classes of the surface's fundamental group. These conjugacy classes can be expressed (not uniquely in the case of closed surfaces) as a cyclic word of minimal length in terms of the fundamental group's generators. The self-intersection number of a conjugacy class is the minimal number of transverse self-intersections of representatives of the class. By using Markov chains to encapsulate the exponential mixing of the geodesic flow and achieve sufficient independence, we can use a form of the central limit theorem to describe the statistical nature of the self-intersection number. For a class chosen at random among all classes of length n, the distribution of the self intersection number approaches a Gaussian when n is large. This theorem generalizes the result of Steven Lalley and Moira Chas to include the case of closed surfaces.

#### Bohr Sets of Integers and Combinatorial Number Theory

When: Tue, November 20, 2012 - 3:30pm
Where: MATH 1308
Speaker: Hillel Furstenberg (Hebrew University, Jerusalem) -
Abstract: Note unusual day, room and time: this talk is given in the Student Dynamics Seminar :
http://www-math.umd.edu/research/seminars/student-dynamics-seminar.html .
It is based on joint work with Vitaly Bergelson and Benjamin Weiss.

#### The Kepler Problem on the Heisenberg Group

When: Thu, November 29, 2012 - 2:00pm
Where: Math 1311
Speaker: Corey Shanbrom, UCSC, cshanbro@ucsc.edu

Abstract: The Kepler problem is among the oldest and most fundamental problems in dynamics. It has been studied in curved geometries, such as the sphere and hyperbolic plane. Here, we formulate the problem on the Heisenberg group, the simplest sub-Riemannian manifold. Key to this formulation is a 1973 result of Folland, who found the fundamental solution to the Heisenberg sub-Laplacian. We will discuss the geometry of this space and present partial results and first steps towards a solution to the Kepler-Heisenberg problem.

#### Cusp Excursions, Random Lattices, and Gap Distributions

When: Mon, December 3, 2012 - 4:15pm
Where: Math 3206 (Colloquium room)

Abstract: Starting from problems of understanding how orbits of
unipotent flows on the space of lattices visit neighborhoods of
infinity, we arrive at a problem on the shape of random unimodular
lattices, and even more unexpectedly a new geometric proof of Hall's
theorem on the gap distribution of Farey fractions, and a proof of a
conjecture of Boca-Zaharescu. This talk includes elements from joint
work with G. Margulis, and joint work with Y. Cheung.

#### On Dettmann's 'Horizon' Conjectures

When: Thu, December 6, 2012 - 2:00pm
Where:
Speaker: Peter Nandori,
Budapest (Hungary), nandori@math.bme.hu

Abstract:
In the simplest case, consider a Z^d-periodic (d >2)
arrangement of balls of radii < 1/2, and select a random direction
and point (outside the balls). According to Dettmann's first
conjecture (Journal of Stat. Phys. 146:181-204, 2012.)
the probability that the so determined free flight (until
the first hitting of a ball) is larger than t >>1 is roughly C/t,
where C is explicitly given by the geometry of the model. In its simplest form,
Dettmann's second conjecture is related to the previous case with
tangent balls (of radii 1/2). The conjectures are established in
quite a general setup: for L-periodic configuration of - possibly
intersecting - convex bodies
with L being a non-degenerate lattice. These questions are related
to P\'olya's visibility problem (1918), to theories of
Bourgain-Golse-Wennberg (1998-) and of
Marklof-Str\"{o}mbergsson (2010-). The results
also provide the asymptotic
covariance of the periodic Lorentz process assuming it has a limit
in the super-diffusive scaling, a fact if d = 2 and the horizon is
infinite. This is a joint work with D. Sz\'asz and T. Varj\'u,
http://arxiv.org/abs/1210.2231.

#### Weakly integrable hamiltonian systems in the two torus and rigidity

When: Thu, January 24, 2013 - 2:00pm
Where: Math 1311
Speaker: Rafael Ruggiero ( Pontificia Universidade Catolica do Rio de Janeiro ) - http://www.mat.puc-rio.br/pagina.php?id=docentes_rruggiero
Abstract: An energy level of a Hamiltonian is called C^{k} integrable if the level admits a C^{k} foliation by invariant, Lagrangian submanifolds. We show that if the unit tangent bundle of a k-basic Finsler metric in the two torus is C^{2} integrable then the flag curvature of the metric is zero. A Finsler metric is called k-basic if the flag curvature does not depend on vertical variables. This result is motivated by the so-called Hopf conjecture that is known to be false in Finsler geometry after Busemann's work, but holds for some Finsler, non-Riemannian metrics called Landsberg metrics.

#### Classification of beta-shifts (joint with Student Seminar)

When: Tue, January 29, 2013 - 11:00am
Where: TBA
Speaker: Rune Johansen (University of Copenhagen)

#### Global observables and infinite mixing (joint with Student Seminar)

When: Tue, January 29, 2013 - 3:30pm
Where: Math 1308
Speaker: Marco Lenci, Universita' di Bologna, Italy

Abstract: Finding a satisfactory definition of mixing for dynamical systems
preserving an infinite measure (in short, infinite mixing) is an
important open problem. Virtually all the definitions that have been
attempted so far use ’local observables’, that is, functions that
essentially only “see” finite portions of the phase space. We
introduce the concept of ’global observable’, a function that gauges a
certain quantity throughout the phase space. This concept is based on
the notion of infinite-volume average, which plays the role of the
expected value of a global observable. Endowed with these notions,
which are to be specified on a case-by-case basis, we give a number of
definitions of infinite mixing. These fall in two categories:
global-global mixing, which expresses the “decorrelation” of two
global observables, and global-local mixing, where a global and a
local observable are considered instead. Time permitting, we will see
how these definitions respond on some examples of
infinite-measure-preserving dynamical systems.

#### Flow equivalence of sofic shifts

When: Thu, January 31, 2013 - 2:00pm
Where: Room 1311
Speaker: Soren Eilers (University of Copenhagen) - http://www.math.ku.dk/~eilers/

Abstract: Franks resolved in 1984 the fundamental question of when two
irreducible shifts of finite type (SFTs) are flow equivalent, offering a
complete invariant which is both easy to compute and easy to compare. In
joint work with Boyle and Carlsen, we have embarked on the task of trying
to similarly classify irreducible sofic shifts up to flow equivalence. It
turns out that solving this problem even in the basic case of AFT shifts involves
understanding the flow classification both of reducible SFTs and of SFTs
with actions of cyclic groups, and I will report on the status of this
project. In the first part of the talk, I will detail the problem, present
a class of simple examples and show how our work allows their classification.
In the second part I will describe how our results are derived from an
extension theorem based on a profound result by Boyle and Krieger.

#### Finite Factor Representations and Totally non-free actions of countable groups

When: Thu, February 7, 2013 - 2:00pm
Where: Math 1311
Speaker: Konstantin Medynets- U.S. Naval Academy

Abstract: We will talk about relations between ergodic properties of group actions and the structure of group characters (the latter is equivalent to the classification of all finite-type factor representations). The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)) for some special group action on a measure space. Here Fix(g) is the set of all fixed points of group element $g$ and \mu is an invariant measure.

We will establish this conjecture for two classes of groups --- (1) full groups of Bratteli diagrams and (2) the Thompson-Higman family of groups. Since the latter class of groups have no non-trivial ergodic measures, we will show that they admit no non-trivial character. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325.

#### Abundance of completely parabolic motions via weak KAM theory

When: Thu, February 14, 2013 - 2:00pm
Where: Math 1311

Abstract: In this talk I will consider the general N-body problem. I will show
a Holder estimate for the critical action potential which enable us
to prove the existence of weak solutions for the Hamilton-Jacobi equation.
The calibrating curves of a given such solution produces a lamination
of the configuration space composed by rays of free time minimizers which,
as we will see, they are completely parabolic motions.

#### ERGODIC PROPERTIES OF SKEW PRODUCTS IN INFINITE MEASURE

When: Thu, February 21, 2013 - 2:00pm
Where: Math 1311
Speaker: Yuri Lima, Weizmann Institute (Israel), yuri.lima@weizmann.ac.il

Abstract: Let $(\Omega;\mu)$ be a shift of finite type with a Markov probability, and
$(Y; \nu)$ a non-atomic standard measure space. For each symbol i of the symbolic space, let $\Phi_i$ be a measure-preserving automorphism of $(Y; \nu)$. We study skew products of the form $(\omega; y) \to ( \sigma(\omega); \Phi_{\omega_0} (y))$, where
$\sigma$ is the shift map on $(\Omega;\mu)$. We prove that, when the skew product is conservative, it is ergodic if and only if the $\Phi_i$'s have no common non-trivial invariant set. In the second part we study the skew product when $\Omega = \{0; 1\}^{\mathbb Z}$, is a Bernoulli measure, and $\Phi_0$, $\Phi_1$ are $\mathbb R$-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to $\sqrt{n}$, and its trajectories satisfy the central, functional central and local limit theorem. Joint work with Patricia Cirilo and Enrique Pujals.

#### On the ergodicity of billiards in polygons

When: Thu, February 28, 2013 - 2:00pm
Where: Math 1311
Speaker: Giovanni Forni-UMD-College Park-gforni@math.umd.edu

Abstract: Billiards in polygons are of two fundamentally different types: rational and non-rational. Rational billiards are pseudo-integrable, that is, their phase space is foliated by invariant surfaces. It was proved by Kerckhoff, Masur and Smillie in 1986 by Teichmuleller theory that the flow on almost all invariant surface is uniquely ergodic. In contrast, the ergodic theory for non-rational billiards is almost non-existent, except for results based on the rational case obtained by fast approximation methods. In this talk we will formulate a new ergodicity criterion for rational billiards (proved in his thesis by R. Trevino) and a generalization to the non-rational case. As a consequence we will formulate an ergodicity theorem for non-rational billiards under a full measure Diophantine condition on the angles (this is work in progress). The proofs are based on methods of elementary complex analysis in one-variable and analysis on Riemannian manifolds. The talk will not assume any knowledge of Teichmueller theory.

#### NO SEMINAR

When: Thu, March 7, 2013 - 2:00pm
Where:

#### Time-changes of distal minimal flows

When: Thu, March 14, 2013 - 2:00pm
Where: MATH 1311
Speaker: Gernot Greschonig ( InstituteVienna University of Technology (TU Vienna), Institute of Discrete Mathematics and Geometry) - http://dmg.tuwien.ac.at/greschg/
Abstract: A result of Egawa states that a time-change of an equicontinuous real compact metric flow is either equicontinuous or topologically weakly mixing. Using Furstenberg's structure theorem we prove that a time-change of a distal minimal real flow is a distal extension of a weakly mixing factor. These results hold remain valid for a generalisation where the time-change cocycle does not establish a homeomorphism of the real line while between the phase spaces there is just a continuous onto map. This setting describes the structure of the topological Mackey action of a real skew product extension of a distal minimal flow.

NOTE: End time is approximate.

#### Recurrence in pairs - known results and open problems

When: Thu, March 28, 2013 - 2:00pm
Where: Math 1311
Speaker: Piotr Oprocha (AGH Univeristy-Poland) -oprocha@agh.edu.pl

Abstract: Consider a dynamical system on a compact metric space. Roughly speaking, a point x is product recurrent if its returns to any open neighborhood of x can by synchronized with returns of any other recurrent point (in any other dynamical system). In other words, x in pair with any other recurrent point y can return simultaneously to their respective neighborhoods (i.e. the pair (x,y) is recurrent). Points x with the above property (the so-called product recurrent points) have been fully characterized many years ago.

If we weaken assumptions on synchronization (e.g. we demand synchronization with recurrent points y from some specified class of dynamical systems), then we can obtain a larger class of admissible points x.

In this talk we will present some recent results and open problems on product recurrence and related topics.

#### Homeomorphic measures on a Cantor set (joint with Student Dynamics)

When: Tue, April 2, 2013 - 3:30pm
Where: MATH 1308
Speaker: Olena Karpel (Institute for Low Temperature Physics, Kharkov, Ukraine) -
Abstract: Two measures \mu and \nu on a topological space X are called homeomorphic if there exists a homeomorphism f of X such that \mu(A) = \nu(f(A)) for every Borel subset A. We are interested in the problem of classification of Borel probability and infinite measures on a Cantor set with respect to a homeomorphism.

For a wide class of probability measures which E. Akin called good, a criterion of being homeomorphic is known. A full non-atomic measure \mu is good if whenever U, V are clopen sets with \mu(U) < \mu(V), there exists a clopen subset W of V such that \mu(W) = \mu(U). For the class of good probability measures, the set S(\mu) of values of measure \mu on all clopen subsets of X is a complete invariant.

We consider ergodic probability and infinite invariant measures for aperiodic substitution dynamical systems. S. Bezuglyi, J.Kwiatkowski, K.Medynets and B.Solomyak showed that these measures can be described as ergodic measures on non-simple stationary Bratteli diagrams invariant with respect to the cofinal (tail) equivalence relation. We also consider a wide class of infinite measures on a Cantor set. We find necessary and sufficient condition for good measures to be homeomorphic. It turns out, that for good infinite measures, the set S(\mu) is not a complete invariant, we find a new invariant which is complete.

#### NO SEMINAR TODAY

When: Thu, April 4, 2013 - 2:00pm
Where: 1311.0
Speaker: MD-PSU DYNAMICS CONFERENCE () - http://dynamics.math.umd.edu/conferences/md13/

#### Moser's normal form and applications to KAM theory [joint with Student Dynamics Seminar]

When: Tue, April 9, 2013 - 3:30pm
Where: MATH 1308
Speaker: Jacques Fejoz (Université Paris-Dauphine) - http://www.ceremade.dauphine.fr/~fejoz/
Abstract: Moser's normal form is a normal form for vector fields in the neighborhood of vector fields possessing a Diophantine quasiperiodic invariant torus. I will describe how to prove the existence and uniqueness of this normal form using a simple inverse function theorem, and show how many some standard and less standard theorems of KAM theory follow from this normal form.

#### TBA (joint Geometry-Dynamics Seminar)

When: Thu, April 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Jeff Danciger (University of Texas, Austin) -

#### Hamiltonian Conley conjecture and Implications for the Reeb flow

When: Thu, April 25, 2013 - 2:00pm
Where: Math 1311

Abstract: I will sketch the proof of the Hamiltonian Conley Conjecture, i.e., the existence of infinitely many periodic orbits for Hamiltonian systems on certain symplectic manifolds. Moreover, I will discuss the notion of a symplectically degenerate maximum, a special periodic orbit used to localize the problem in the proof of the Conley conjecture.
For the Reeb flow in contact geometry, the existence of a contact analog of a symplectically degenerate maximum implies that there are infinitely many periodic Reeb orbits on a certain class of contact manifolds.
This theorem also implies that the Reeb vector field of every contact form supporting the standard contact structure on the three-sphere has at least two periodic orbits.
The contact part of the talk is joint work with V. Ginzburg, U. Hryniewicz and L. Macarini.

#### The empty talk

When: Thu, May 2, 2013 - 2:00pm
Where: MATH 1311
Speaker: No speaker today () -

#### A report on the solution of Plateau's Problem (soap bubbles)

When: Thu, May 9, 2013 - 2:00pm
Where: MATH 1311
Speaker: Jim Yorke (University of Maryland ) - http://www.chaos.umd.edu/~yorke/