- Date
- 08/08/2014 - 08/12/2014
- Location
- CNU

PAULO BRANDAO
Finiteness of Attractors for Maps of the Interval Allowing Discontinuities
We discuss some new results on Lorenz maps: Contracting Lorenz maps with negative Schwarzian derivative have only one topological attractor. More precisely, if such a map $f$ has no finite attractor (periodic attractors or super-attractors), then there is a transitive compact set $\Lambda$ such that it is the $\omega$-limit set under $f$ for a residual set of the interval. Also, we develop a Spectral Decomposition Theory for Lorenz maps.
We also discuss a joint work with Palis and Pinheiro where we show that a piecewise $C^3$ one-dimensional map with negative Schwarzian derivative and with a finite number of criticalities/discontinuities display only finitely many attractors.

Vilton Pinheiro Measures with historic behavior We say that a point $x$ (or its orbit $\mathcal{O}^{+}(x)$) has historic behavior if $$\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n-1}\varphi(f^{j}(x))$$ does not exist for some $\varphi\in C^{0}(\XX)$. That is, \begin{center} $\frac{1}{n}\sum_{j=1}^{n-1}\delta_{f^{j}(x)}$ does not converges in the weak topology. \end{center} This terminology was introduced by Ruelle in 2001. We will use Caratheodory measures to study points with historic behavior, relating those points to hyperbolicity or homoclinc tangencies.

Masayuki Asaoka Arbitrary growth of the number of periodic orbits in one-dimensional semigroups In 2000, Kaloshin proved that arbitrary growth of the number of periodic orbits is generic in a Newhouse domain for surface diffeomorphisms. All known results on the arbitrary growth is based on homoclinic tangency. So, it is natural to ask whether the arbitrary growth is possible in a open set of partially hyperbolic systems with one-dimensional center, or not. In this talk, we will discuss the corresponding problem for one-dimensional semigroups, which is a model of the dynamics along the cnter direction of a partially hyperbolic system. In fact, we will show that arbitrary growth of the number of periodic orbit is generic in any open set in which semigroups admit a blender and heteroclinic orbits with a mild condition. This is a joint work with K.Shinohara and D.Turaev.

Anna Dorjieva The computational technology for searching limit cycles of the dynamical systems We formulate the problem of finding limit cycles for dynamical systems as global optimization problem. For solving this problem we propose the OPTCON-A software to apply. In the talk we discuss results of numerical experiments on the test collection of such problems.

Alexander Gornov Technology of computational investigating of non-convex optimal control problems in systems that are not solved for the derivatives We propose a set of reductions, allowing to reduce the initial extreme models in the form of ordinary differential equations that are not solved for the derivative to optimal control problems in the system of differential equations in the Cauchy form. It is carried out the numerical experiments to study the properties of proposed reductions in the problem of stabilization of the pendulum in the linear and nonlinear formulation and others. We use software OPTCON-A for solving considered optimization problems. Implemented computational experiments have demonstrated the effectiveness of the proposed approach.

Nguyen Huu Du Some recent results in differential equations perturbed by noise \begin{document} \title{\bf Some recent results in differential equations perturbed by noise} \maketitle \centerline{\bf N.H. Du, N.H. Dang and N. T. Hieu} \vskip .1cm \centerline{VNU-Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Ha Noi} \vskip .1cm \centerline{\bf N. T. Dieu} \centerline{ Vinh University, 182 Le Duan, Vinh Nghe An } In this talk, we deal with the dynamic behavior of Kolmogorov systems perturbed by telegraph noise: {e1.1} x = x a(t,\xi(t),x,y) \\ y = y b(t,\xi(t),x,y), where, $(\xi(t))$ is a telegraph noise with valued on a set of two elements, say $E=\{-,+\}$, $ a,\; b $ two functions. \item[2)] Or, systems perturbed by white noise with delay \begin{equation*}\label{d0.1} dX(t) = f\big(X(t), X(t - \tau), \xi(t)\big) dt + g\big(X(t), X(t - \tau), r(t)\big) dB(t), \end{equation*} here $$f:\R^n\times\R^n\times S\to\R^n,\qquad g:\R^n\times\R^n\times S\to\R^n$$ and $B(t)$ is Brownian motion. \end{enumerate} We are interested in the description of the $\omega-$ limit set of each solution, the attractor and also the stability in distribution of these systems.

Antoine Gomis Riemann Hypothesis, Number Theory and Dynamics It is shown that the Riemann Hypothesis is equivalent to the statement that its complex zeros are also zeros of a canonical quadratic equation. This equation commands a dynamic system that will be outlined. The implications of this Riemann dynamics will be developed.

Xiangdong Ye Topological models and the pointwise convergence Strengthening the well-known Jewett-Krieger's Theorem by requiring that it is strictly ergodic under some group actions, and building the connection of the new model with the pointwise convergence of non-conventional ergodic averages we prove that for an ergodic system $(X,\X,\mu, T)$, $d\in\N$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, and any tempered F{\rm ${\o}$}lner sequence $\{F_N\}_{N\ge 1}$ of $\Z^2$, the averages \frac{1}{|F_N|} \sum_{(n,m)\in F_N} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) converge $\mu$ a.e. We remark that the same method can be used to show the pointwise convergence of ergodic averages along cubes which was firstly proved by Assani and then extended to a general case by Chu and Franzikinakis. This is a joint work wit Huang and Shao.

Hyekyoung Choi On the flips for a synchronized system (To appear in extit{Erogodic theory and dynamical systems}) In this talk, we will be interested in the following property for a shift space $X$: If $X$ has a flip, then it has infinitely many non-conjugate ones. We show that every synchrnozied system has the property, but there is a coded system which does not have the property. We recall that every irreducible sofic shift is a synchronized system and every synchronized system is a coded system. We construct a coded system which has a flip $phi$, but every flip for the coded system is conjugate to one of the two flips $phi$ and $sigmaphi$. Thus the coded system does not have the property. The construction relies on a method developed by D. Fiebig and U. Fiebig. This is a joint work with Young-One Kim.

Dmitry Todorov Lipschitz inverse shadowing and structural stability There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let $M$ be a closed Riemannian manifold. Diffeomorphism $f:M o M$ is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. A similar definition can be given for flows. Most results about this property prove that it is present in certain hyperbolic situations. Quite surprisingly, recently it has been proven that a quantitative version of it is in fact equivalent to hyperbolicity (structural stability). There is also a notion of emph{inverse} shadowing that is a kind of a converse to the notion of classical shadowing. Dynamical system is said to have inverse shadowing property if for any (exact) trajectory there exists a pseudotrajectory from a special class that is uniformly close to the original exact one. I will describe a quantitative (Lipschitz) version of this property and why it is equivalent to structural stability both for diffeomorphisms and for flows.

Soonjo Hong More results on class degrees of factor maps TBA

Uijin Jung Class degree and the structure of transition classes: a new invariant for infinite-to-one maps from shifts of finite type It is well known that given finite-to-one factor code from an irreducible shift of finite type, almost all points have the same number of inverse images. This number is called the degree of a code, and widely studied and proved useful in the study of entropy-preserving factor maps between shifts of finite type and those of sofic shifts. The class degree is a new invariant for infinite-to-one codes (i.e., non entropy-preserving maps) defined in terms of transition classes. In this talk, we investigate structure of transition classes and present dynamical properties of transition classes analogous to the properties of fibers of finite-to-one codes. This is a joint work with Mahsa Allahbakhshi and Soonjo Hong (Centro de Modelamiento Matemático, Universidad de Chile)

Shuhei Hayashi A refinement of Ma¥~n¥'e's $C^1$ generic dichotomy We consider a refinement of Ma¥~n¥'e's $C^1$ generic dichotomy for surface diffeomorphisms, that is, $C^1$ generically they have either (I) hyperbolicity; or (II) (by taking the inverse if necessary) infinitely many attracting periodic orbits each of which has either relatively large immediate basins of attraction (observability) or a pathological feature. This is a consequence of general perturbation result creating sinks with relatively large immediate basins of attraction under the existence of a non-atomic ergodic measure admitting at most small positive Lyapunov exponents.

Yingfei Yi TBA The talk will start from the origin of ergodic theory and discuss connections among three different approaches in describing isolated physical systems at microscopic (Hamiltonian), macroscopic (statistical), or mesoscopic (mixed) levels. Characterization of limiting Gibb's measures will be given in responding to the desired ergodicity under thermodynamics limits. Some general theory on the existence, concentration, and limiting behavior of stationary measures of a diffusion process will also be presented.

Matthew Nicol Limit laws for sequential and non-stationary dynamical systems We establish probabilistic limit laws such as the central limit theorem, almost sure invariance principle and dynamical Borel-Cantelli lemmas for certain sequential and other non-stationary dynamical systems. A sequential dynamical system refers to a non-stationary sequence of maps $T_k\circ T_{k-1}\circ T_1$ acting on a space $X$. The maps are allowed to vary with $i$.

Yinghao Han Attractors for a Wave Equation on R3 with Linear Memory We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity on $R^3$ $$ u_{tt}-k(0)Delta u-int_{0}^{infty}k^{'}{(s)}Delta u {(t-s)}ds+g(u)=f(x). onumber $$ In which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. In the autonomous case, the existence of a attractor is achieved. The result applies to the so-called gradient systems, that is, dynamical systems possessing a global Lyapunov functional.

Maria José Pacifico Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincar\'{e} maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.

NGAMENI NGAMCHO FAUSTIN OLATE MONSIEUR WE NEED TO COME FOR SEE NEW PARTENAR

Dibyendu De Multiplicatively Large Sets in $Z[i]$ In the recent century one of the celebrated theorem in additive combinatorics is Green-Tao Theorem: ``primes contain arbitrary long arithmatic progressions''. This theorem comes as a particular case of Erdos conjecture: if $A$ be a subset of $\mathbb{N}$ with the property that ${\displaystyle \sum_{n\in A}\frac{1}{n}}\rightarrow\infty$ then $A$ contains arithmatic progressions arbitrary length. Green-Tao Theorem is greatly indebt to to Furstenberg ergodic theoritic proof of Szameredi's Theorem. This Theorem states that every subset of $mathbb{N}$ with positive upper banach density of contains contains arbitrary long arithmatic progressions. Szameredi's Theorem was purly combinatorial and involves sophisticated graph theoretic method. Furstenberg translated Szameredi's proof in ergodic theoretic set up. Tao and Green, in their proof used a concept of ``positive relative density'' with respect to primes. Main invention in their proof is to introduce so called pseudo random measure. In 2006 Terence Tao himself extended Green-Tao Theorem for the integral domain $mathbb{Z}[i]$, the set of Gaussian integers. In 2010 Tao's scholar Thai Hoang Le extended Green-Tao Theorem for function fields $\mathbb{\mathbb{F}}_{q}[x]$. Furstenberg ergodic theoritic proof of Szameredi's Theorem was so powerful that it opened a new branch in research, called ``Ergodic Ramsey Theory''. Vitaly Bergelson was the first person who investigated combinatorial structures of subsets of $\mathbb{N}$ with positive multiplicative density. Using various ergodic multiple recurrence theorems, Bergelson proved that multiplicatively large sets i.e. sets with positive multiplicative density have a rich combinatorial structure. He proved that for any multiplicatively large set $A\subset\mathbb{N}$ and any $k\in\mathbb{N}$, there exists $a,b,c,d,e,q\in\mathbb{N}$ such that $\{q^{j}(a+id):0\leq i,j\leq k\}\subset A$ and $\{b(c+ie)^{j}:0\leq i,j\leq k\}\subset A$ In this presentation we to extend these results for integral domains $\mathbb{Z}[i]$.

Stefanie Hittmeyer Bifurcations of generalised Julia sets near the complex quadratic family We consider a two-dimensional noninvertible map that was introduced by Bamon, Kiwi and Rivera in 2006 as a model of wild Lorenz-like chaos [1,2]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimage. The bounding critical circle and its images, together with the critical point and its preimages, form the critical set. In a specific parameter regime the map is a nonanalytic perturbation of the complex quadratic family. As parameters are varied away from the complex quadratic family the dynamics on the plane initially stay qualitatively the same. On the other hand, saddle points and their stable and unstable sets then appear as new ingredients of the dynamics. The stable, unstable and critical sets interact with the generalised Julia set, leading to the (dis)appearance of chaotic attractors and to dramatic changes in the topology of the generalised Julia set [3]. In particular, we find generalised Julia sets in the form of Cantor bouquets, Cantor tangles and Cantor cheeses. Continuation of these bifurcations in two parameters reveals a self-similar bifurcation structure near the period-doubling route to chaos of the complex quadratic family. [1] R. Bamon, J. Kiwi, and J. Rivera, Wild Lorenz-like attractors, Preprint, 2006, arXiv:math0508045v2 [2] S. Hittmeyer, B. Krauskopf and H.M. Osinga, Interacting global invariant sets in a planar map model of wild chaos, SIAM J. .Appl. Dyn. Syst., 12(3), 2013 [3] S. Hittmeyer, B. Krauskopf and H.M. Osinga, Interactions of the Julia set with critical and (un)stable sets in an angle-doubling map on C\{0}, Preprint, 2013

Dong Han Kim Subword complexity and Sturmian colorings of regular trees In this talk, we introduce subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and then introduce Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ``infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones. We will provide several examples. This is joint work with Seonhee Lim.

Bing Li Intermediate beta-shifts of finite type An aim of this talk is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\beta,\alpha} \colon x \mapsto \beta x + \alpha \bmod 1$ for which the corresponding intermediate $\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\beta,\alpha)$ such that the intermediate $\beta$-shift associated with $T_{\beta, \alpha}$ is a subshift of finite type and where the map $T_{\beta,\alpha}$ is not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\beta$-shifts and $\beta$-transformations, in that these two properties do not hold.

Yi-Chiuan Chen TBA Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.

Elisabete Alberdi Celaya Object Oriented Programming methodology for Dynamical Finite Element Method Computations E. ALBERDI CELAYA and J. J. ANZA Numerous phenomena of science and engineering are modelled mathematically using systems of Partial Derivative Equations (PDEs). The analytical solution of PDEs in a general domain is not possible and it is necessary to use numerical methods, among which the Finite Difference Method, the Boundary Element Method and the Finite Element Method (FEM) stand out, being the FEM the most capable in general, to deal with any shape domains and non linear problems. Traditionally, the development of numerical software has been based on the use of procedural languages, but in the last years there is an increasing interest of applying the paradigms of the Object Oriented Programming (OOP), which allows an efficient reutilization, extension and maintenance of the codes. After a short introduction to the mathematical modelling of the continuous media, which shows the similarity between the governing Partial Differential Equations in different applications, common blocks for Finite Element approximation are identified, and an Object Oriented Programming (OOP) methodology for linear and non linear, stationary and dynamic problems is presented. Advantages of this approach are commented and some results are shown as examples of this methodology. References: 1.- . Belytschko, W. K, Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons, Chichester, 2000. 2.- J. Bonet, R. D. Wood, Non linear continuum mechanics for finite element analysis, Cambridge University Press, Cambridge, 1997. 3.- K. F. Graff, Wave motion in elastic solids, Oxford University Press, London, 1975. 4.- E. Hairer, G. Wanner, Solving ordinary differential equations, II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996. 5.- T. J. R. Hughes, The Finite Element Method. Linear static and dynamic analysis, Prentice Hall, New Jersey, 1987. 6.- H. P. Langtangen, Computational Partial Differential Equations- Numerical Methods and Diffpack Programming, Springer-Verlag, Berlin, 1999. 7.- J. C. Simo, T. J. R. Hugues, Computational Inelasticity, Springer Verlag, New York, 1997.

Hoang Duc Luu Local metric entropy for finite time nonautonomous dynamical systems We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesin's entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special dynamical behavior such as stable and unstable manifolds and Lagrangian coherent structures.

Wenxiang Sun Degeneracy of entropy of flow with singurarity While entropy measures dynamical complexity and an invariant for disrcete systems, entropy is degenerate for flows with singurarity: there exists an equivalent pair of flows with singularity such that one has zero entropy and the other has infinite entropy. Such entropy degeneracy happens if and only if all invariant ergodic measures with positive entropy near singularities loss their invarance, when shiftting equivalently one flow to another.

Sergey Tikhomirov Shadowing in actions of non-abelian groups We introduce notion of shadowing property for actions of finitely generated not necessarily abelian groups. In contract with shadowing for diffeomorphisms and flows we show that shadowing property depends not only on hyperbolicity but on the group structure as well. For nilpotent groups we prove analog of shadowing lemma. We give an example of an action of a solvable group, whose shadowing property depends on quantitative properties of hyperbolicity. Finally we prove that there is no linear action of free nonabelian group which has shadowing property.

Sergei Pilyugin Shadowing and structural stability of diffeomorphisms and flows We discuss several recent results on relations between shadowing and structural stability of diffeomorphisms and flows.

Alba Marina Málaga Sabogal Generic minimality and almost sure conservativity for a family of dynamical systems on the cylinder S¹×Z I work on a family of (discrete) dynamical systems which is heuristically related to a billiard on a parallelogram. This family is defined on the discrete cylinder $\mathbb S^1\times \mathbb Z$ where $\mathbb T^1={\mathbb R/\mathbb Z}$ is the one-dimensional torus (i.e. the circle). For any bi-infinite sequence $\underline\alpha\in\mathbb T^\mathbb Z$, we define the transformation $F_{\underline\alpha}$ almost everywhere on the cylinder as follows:$$F_{\underline\alpha}\left([x]_\mathbb Z,n\right)=\left([x+\alpha_n]_\mathbb Z, n+\left\{\begin{array}{rcl}1&if &x+\alpha_n\in(0,\frac12)+\mathbb Z\\-1&if&x+\alpha_n\in(-\frac12,0)+\mathbb Z\\ \end{array}\right.\right).$$When the sequence $\underline\alpha$ is constant and irrational, Conze and Keane showed in {[1]} that $F_{\underline\alpha}$ is ergodic.\\I am trying to understand what are the typical properties of $F_{\underline\alpha}$ in the following meaning. Namely, what properties hold for almost any $\underline\alpha$ or for a generic $\underline\alpha$ in the parameter space? For the moment I have proved that conservativity is both generic and almost-sure, whereas the minimality is generic. I would like to understand also the diffusion properties of this family. Bibliography Conze, J.-P.; Keane, M. Ergodicité d'un flot cylindrique. (French) Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976), Exp. No. 5, 7 pp. Dépt. Math. Informat., Univ. Rennes, Rennes, 1976.

Elisabete Alberdi Celaya Application of some numerical damping control multistep methods E. ALBERDI CELAYA and J.J. ANZA AGUIRREZABALA The second order Ordinary Differential Equation (ODE) system obtained after semidiscretizing the wave-type partial differential equation (PDE) with the finite element method (FEM) shows strong numerical stiffness. Its resolution requires the use of numerical methods with good stability properties and controlled numerical dissipation in the high-frequency range. Some of the methods developed in the linear range are the Collocation method, the Wilson method, the HHT-$alpha$ method, the Houbolt method, or more recent works as the generalized-alfa method. The development of similar methods for nonlinear problems is more recent and it has been originated by the presence of numerical instabilities when solving nonlinear problems by methods which are unconditionally stable in the linear range. In this paper, we work in the linear range and we formulate a new method called BDF-$alpha$, which is second-order accurate, unconditionally stable for some values of the parameter $alpha$ and permits a parametric control of numerical dissipation. Some applications of this new method are shown. References: 1.- F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for nonlinear dynamics. Part I: low order methods for two model problems and nonlinear elastodynamics, Comput. Meth. Appl. Mech. Eng., Vol.190, 2603-2649, 2000. 2.- F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for nonlinear dynamics. Part II: second order methods, Comput. Meth. Appl. Mech. Eng., Vol.190, 6783-6824, 2001. 3.- J. Chung, G. M. Hulbert, A time integration algorithm for structural dynamics with improvednumerical dissipation: the generalized-$alpha$ method, Journal of Applied Mechanics, Vol.60, 371-375, 1993. 4.- O. Gonzalez, Exact energy-momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Eng., Vol.190, 1763-1783, 2000. 5.- E. Hairer, G. Wanner, Solving ordinary differential equations, II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996. 6.- H. M. Hilber, T. J. R. Hughes, Robert L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn., Vol.5, 283-292, 1977. 7.- T. J. R. Hughes, The finite element method. Linear Static and dynamic finite element analysis, Prentice-Hall International Editions, New Jersey, 1987.

Piotr Oprocha Proximal and completely scrambled systems A dynamical system $(X,f)$ is completely scrambled if every pair $x,y$ of distinct points in $X$ is proximal (i.e. $\liminf d(f^n(x),f^n(y))=0$) but not asymptotic (i.e. $\limsup d(f^n(x),f^n(y))>0$). In 1999, Huang and Ye proved that continua of arbitrary dimension admits completely scrambled homeomorphism. In this talk we aim to present a brief characterization of completely scrambled homeomorphisms, ideas behind the construction of Huang and Ye and some other methods of construction of completely scrambled systems which are in fact motivated by much earlier papers (on a different topic). If time permits, we will present a method to answer a question left open in the Huang and Ye's paper (joint work with Fory\'s, Huang and Li). Finally, we will present a few questions for further research.

Christian Rodrigues TBA Amongst the main concerns of Dynamics one wants to decide whether asymptotic states are robust under random perturbations. Considering the iteration of $f : M \to M$, such randomness is implemented by a family $\{p_{\varepsilon}(\, \cdot \,|x)\}$ of Borel probability measures, such that every $p_{\varepsilon}(\, \cdot \,|x)$ is supported inside the $\varepsilon$-neighbourhood of $f(x)$. Alternatively, the orbit is given by the iteration $x_{j} = g_{j} \circ \cdots \circ g_{1}(x_{0})$, where each measurable $g_{j}$ is picked at random $\varepsilon$-close from the original map $f$. Endowing the collection of maps $\{g_{j}\}$ with a probability distribution $\nu_{\varepsilon}$, we say that the sequence of random maps is a representation of that Markov chain if for every Borel subset $U$, $p_{\varepsilon}(U|x) = \nu_{\varepsilon}(\{g : g(x) \in U \})$. In this talk we systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. From this scheme, we not only deduce the representation by measurable and continuous random maps, but also obtain conditions for the to construct random diffeomorphisms from a given Markov chain. This is a joint work with Jost, and Kell.

Dawei Yang Typical dynamics of three-dimensional I will talk about what are the typical dynamics of the flows generated by three-dimensional vector fields. And I will discuss how to get singular hyperbolicity from domination.

Peter Kloeden Semi-Hyperbolicity and Bishadowing Semi-hyperbolicity extends the classical concept of hyperbolicity for diffeomorphisms to noninvertible Lipschitz mappings on sets that need not be invariant. Moreover, the stable and unstable manifolds need to be mapped only approximately onto their counterparts. Semi-hyperbolic mappings are bi-shadowing, i.e., they also satisfy a converse form of shadowing in which the pseudo-orbits are not arbitrary but are the true orbits of admissible classes of approximating mappings. The theory of these concepts is given and illustrated through applications of nonsmooth systems including those with delays and hysteresis, as well as numerical approximations.

In-Jee Jeong Outer Billiards with Contraction We consider the outer billiards map around a convex polygon, but now composed with a linear contraction by $lambda$. Experiments suggest that for almost every polygons and values of $lambda$, there are only finitely many periodic orbits to which all orbits are attracted. We prove these "regularity" results in the case when the polygon is either a triangle, parallelogram, or a regular hexagon. On the other hand, it turns out that for every quadrilateral, there is a special value of $lambda$ where the dynamics is reduced to that of a piecewise contractive map of the interval. From this, we can show that for certain quadrilaterals there is an attracting Cantor set. The ergodic properties will be discussed as well.

Nandadulal Bairagi Nonlinear dynamics in relation to mathematical ecology and epidemiology TBA

Valery Gaiko Global limit cycle bifurcations in polynomial dynamical systems The global qualitative analysis of polynomial dynamical systems is carried out. Using a new bifurcational geometric method, we solve the problem of the maximum number of limit cycles surrounding a singular point for an arbitrary planar polynomial system and Hilbert's Sixteenth Problem for the general Lienard polynomial system with an arbitrary (but finite) number of singular points. Applying this method, we study also three-dimensional polynomial systems and, in particular, complete the strange attractor bifurcation scenario in the classical Lorenz system connecting globally the homoclinic, period-doubling, Andronov-Shilnikov, and period-halving bifurcations of its limit cycles.

Dominik Kwietniak Specification-like properties and invariant measures I would like to present a survey of various properties similar to Bowen's specification property. In a joint work with Katrin Gelfert we proposed yet another generalization, which is strong enough to imply that the set of ergodic measures is dense in the simplex of all invariant measures and every invariant measure has a generic point. I will describe our approach and provide examples which allow to distinguish between our approach and old and recent specification-like methods of Sigmund, Bowen, Climenhaga-Thompson, Pfister-Sullivan.

Edson Vargas Invariant measures for Cherry flows We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the casewhen the saddle is dissipative or conservative we showthat the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle. Joint work with Radu Saghin.

CA Morales Sectional-hyperbolic systems We review progress in the theory of sectional-hyperbolic dynamics.

Rafael POTRIE Quasi-attractors for 3-dimensional diffeomorphisms far from tangencies We discuss finiteness and uniqueness results of quasi-attractors in several contexts for diffeomorphisms of 3-manifolds which are far from homoclinic tangencies. This builds on colaborations with S. Crovisier, A. Hammerlindl, M. Sambarino.

Shin Kiriki Blenders in center unstable H'enon-like families: with an application to heterodimensional bifurcations We give an explicit family of polynomial maps called center unstable H'enon-like maps and prove that they exhibits blenders for some parameter values. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle. This is a joint work with L. J. Díaz and K. Shinohara, to appear in Nonlinearity.

Khosro Tajbakhsh Asymptotic Measure Expansive Diffeomorphisms In this brief paper the notion of asymptotic measure expansiveness and its relationship with the dominated splitting is considered. It is proved that if a diffeomorphism $f$ admits a dominated splitting then it is asymptotic measure expansive. Also, a diffeomorphism with a homoclinic tangency perturbed is reaching to a non-asymptotic measure expansive diffeomorphism.

Vilton Pinheiro Measures with historic behavior We say that a point $x$ (or its orbit $\mathcal{O}^{+}(x)$) has historic behavior if $$\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n-1}\varphi(f^{j}(x))$$ does not exist for some $\varphi\in C^{0}(\XX)$. That is, \begin{center} $\frac{1}{n}\sum_{j=1}^{n-1}\delta_{f^{j}(x)}$ does not converges in the weak topology. \end{center} This terminology was introduced by Ruelle in 2001. We will use Caratheodory measures to study points with historic behavior, relating those points to hyperbolicity or homoclinc tangencies.

Masayuki Asaoka Arbitrary growth of the number of periodic orbits in one-dimensional semigroups In 2000, Kaloshin proved that arbitrary growth of the number of periodic orbits is generic in a Newhouse domain for surface diffeomorphisms. All known results on the arbitrary growth is based on homoclinic tangency. So, it is natural to ask whether the arbitrary growth is possible in a open set of partially hyperbolic systems with one-dimensional center, or not. In this talk, we will discuss the corresponding problem for one-dimensional semigroups, which is a model of the dynamics along the cnter direction of a partially hyperbolic system. In fact, we will show that arbitrary growth of the number of periodic orbit is generic in any open set in which semigroups admit a blender and heteroclinic orbits with a mild condition. This is a joint work with K.Shinohara and D.Turaev.

Anna Dorjieva The computational technology for searching limit cycles of the dynamical systems We formulate the problem of finding limit cycles for dynamical systems as global optimization problem. For solving this problem we propose the OPTCON-A software to apply. In the talk we discuss results of numerical experiments on the test collection of such problems.

Alexander Gornov Technology of computational investigating of non-convex optimal control problems in systems that are not solved for the derivatives We propose a set of reductions, allowing to reduce the initial extreme models in the form of ordinary differential equations that are not solved for the derivative to optimal control problems in the system of differential equations in the Cauchy form. It is carried out the numerical experiments to study the properties of proposed reductions in the problem of stabilization of the pendulum in the linear and nonlinear formulation and others. We use software OPTCON-A for solving considered optimization problems. Implemented computational experiments have demonstrated the effectiveness of the proposed approach.

Nguyen Huu Du Some recent results in differential equations perturbed by noise \begin{document} \title{\bf Some recent results in differential equations perturbed by noise} \maketitle \centerline{\bf N.H. Du, N.H. Dang and N. T. Hieu} \vskip .1cm \centerline{VNU-Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Ha Noi} \vskip .1cm \centerline{\bf N. T. Dieu} \centerline{ Vinh University, 182 Le Duan, Vinh Nghe An } In this talk, we deal with the dynamic behavior of Kolmogorov systems perturbed by telegraph noise: {e1.1} x = x a(t,\xi(t),x,y) \\ y = y b(t,\xi(t),x,y), where, $(\xi(t))$ is a telegraph noise with valued on a set of two elements, say $E=\{-,+\}$, $ a,\; b $ two functions. \item[2)] Or, systems perturbed by white noise with delay \begin{equation*}\label{d0.1} dX(t) = f\big(X(t), X(t - \tau), \xi(t)\big) dt + g\big(X(t), X(t - \tau), r(t)\big) dB(t), \end{equation*} here $$f:\R^n\times\R^n\times S\to\R^n,\qquad g:\R^n\times\R^n\times S\to\R^n$$ and $B(t)$ is Brownian motion. \end{enumerate} We are interested in the description of the $\omega-$ limit set of each solution, the attractor and also the stability in distribution of these systems.

Antoine Gomis Riemann Hypothesis, Number Theory and Dynamics It is shown that the Riemann Hypothesis is equivalent to the statement that its complex zeros are also zeros of a canonical quadratic equation. This equation commands a dynamic system that will be outlined. The implications of this Riemann dynamics will be developed.

Xiangdong Ye Topological models and the pointwise convergence Strengthening the well-known Jewett-Krieger's Theorem by requiring that it is strictly ergodic under some group actions, and building the connection of the new model with the pointwise convergence of non-conventional ergodic averages we prove that for an ergodic system $(X,\X,\mu, T)$, $d\in\N$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, and any tempered F{\rm ${\o}$}lner sequence $\{F_N\}_{N\ge 1}$ of $\Z^2$, the averages \frac{1}{|F_N|} \sum_{(n,m)\in F_N} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) converge $\mu$ a.e. We remark that the same method can be used to show the pointwise convergence of ergodic averages along cubes which was firstly proved by Assani and then extended to a general case by Chu and Franzikinakis. This is a joint work wit Huang and Shao.

Hyekyoung Choi On the flips for a synchronized system (To appear in extit{Erogodic theory and dynamical systems}) In this talk, we will be interested in the following property for a shift space $X$: If $X$ has a flip, then it has infinitely many non-conjugate ones. We show that every synchrnozied system has the property, but there is a coded system which does not have the property. We recall that every irreducible sofic shift is a synchronized system and every synchronized system is a coded system. We construct a coded system which has a flip $phi$, but every flip for the coded system is conjugate to one of the two flips $phi$ and $sigmaphi$. Thus the coded system does not have the property. The construction relies on a method developed by D. Fiebig and U. Fiebig. This is a joint work with Young-One Kim.

Dmitry Todorov Lipschitz inverse shadowing and structural stability There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let $M$ be a closed Riemannian manifold. Diffeomorphism $f:M o M$ is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. A similar definition can be given for flows. Most results about this property prove that it is present in certain hyperbolic situations. Quite surprisingly, recently it has been proven that a quantitative version of it is in fact equivalent to hyperbolicity (structural stability). There is also a notion of emph{inverse} shadowing that is a kind of a converse to the notion of classical shadowing. Dynamical system is said to have inverse shadowing property if for any (exact) trajectory there exists a pseudotrajectory from a special class that is uniformly close to the original exact one. I will describe a quantitative (Lipschitz) version of this property and why it is equivalent to structural stability both for diffeomorphisms and for flows.

Soonjo Hong More results on class degrees of factor maps TBA

Uijin Jung Class degree and the structure of transition classes: a new invariant for infinite-to-one maps from shifts of finite type It is well known that given finite-to-one factor code from an irreducible shift of finite type, almost all points have the same number of inverse images. This number is called the degree of a code, and widely studied and proved useful in the study of entropy-preserving factor maps between shifts of finite type and those of sofic shifts. The class degree is a new invariant for infinite-to-one codes (i.e., non entropy-preserving maps) defined in terms of transition classes. In this talk, we investigate structure of transition classes and present dynamical properties of transition classes analogous to the properties of fibers of finite-to-one codes. This is a joint work with Mahsa Allahbakhshi and Soonjo Hong (Centro de Modelamiento Matemático, Universidad de Chile)

Shuhei Hayashi A refinement of Ma¥~n¥'e's $C^1$ generic dichotomy We consider a refinement of Ma¥~n¥'e's $C^1$ generic dichotomy for surface diffeomorphisms, that is, $C^1$ generically they have either (I) hyperbolicity; or (II) (by taking the inverse if necessary) infinitely many attracting periodic orbits each of which has either relatively large immediate basins of attraction (observability) or a pathological feature. This is a consequence of general perturbation result creating sinks with relatively large immediate basins of attraction under the existence of a non-atomic ergodic measure admitting at most small positive Lyapunov exponents.

Yingfei Yi TBA The talk will start from the origin of ergodic theory and discuss connections among three different approaches in describing isolated physical systems at microscopic (Hamiltonian), macroscopic (statistical), or mesoscopic (mixed) levels. Characterization of limiting Gibb's measures will be given in responding to the desired ergodicity under thermodynamics limits. Some general theory on the existence, concentration, and limiting behavior of stationary measures of a diffusion process will also be presented.

Matthew Nicol Limit laws for sequential and non-stationary dynamical systems We establish probabilistic limit laws such as the central limit theorem, almost sure invariance principle and dynamical Borel-Cantelli lemmas for certain sequential and other non-stationary dynamical systems. A sequential dynamical system refers to a non-stationary sequence of maps $T_k\circ T_{k-1}\circ T_1$ acting on a space $X$. The maps are allowed to vary with $i$.

Yinghao Han Attractors for a Wave Equation on R3 with Linear Memory We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity on $R^3$ $$ u_{tt}-k(0)Delta u-int_{0}^{infty}k^{'}{(s)}Delta u {(t-s)}ds+g(u)=f(x). onumber $$ In which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. In the autonomous case, the existence of a attractor is achieved. The result applies to the so-called gradient systems, that is, dynamical systems possessing a global Lyapunov functional.

Maria José Pacifico Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincar\'{e} maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.

NGAMENI NGAMCHO FAUSTIN OLATE MONSIEUR WE NEED TO COME FOR SEE NEW PARTENAR

Dibyendu De Multiplicatively Large Sets in $Z[i]$ In the recent century one of the celebrated theorem in additive combinatorics is Green-Tao Theorem: ``primes contain arbitrary long arithmatic progressions''. This theorem comes as a particular case of Erdos conjecture: if $A$ be a subset of $\mathbb{N}$ with the property that ${\displaystyle \sum_{n\in A}\frac{1}{n}}\rightarrow\infty$ then $A$ contains arithmatic progressions arbitrary length. Green-Tao Theorem is greatly indebt to to Furstenberg ergodic theoritic proof of Szameredi's Theorem. This Theorem states that every subset of $mathbb{N}$ with positive upper banach density of contains contains arbitrary long arithmatic progressions. Szameredi's Theorem was purly combinatorial and involves sophisticated graph theoretic method. Furstenberg translated Szameredi's proof in ergodic theoretic set up. Tao and Green, in their proof used a concept of ``positive relative density'' with respect to primes. Main invention in their proof is to introduce so called pseudo random measure. In 2006 Terence Tao himself extended Green-Tao Theorem for the integral domain $mathbb{Z}[i]$, the set of Gaussian integers. In 2010 Tao's scholar Thai Hoang Le extended Green-Tao Theorem for function fields $\mathbb{\mathbb{F}}_{q}[x]$. Furstenberg ergodic theoritic proof of Szameredi's Theorem was so powerful that it opened a new branch in research, called ``Ergodic Ramsey Theory''. Vitaly Bergelson was the first person who investigated combinatorial structures of subsets of $\mathbb{N}$ with positive multiplicative density. Using various ergodic multiple recurrence theorems, Bergelson proved that multiplicatively large sets i.e. sets with positive multiplicative density have a rich combinatorial structure. He proved that for any multiplicatively large set $A\subset\mathbb{N}$ and any $k\in\mathbb{N}$, there exists $a,b,c,d,e,q\in\mathbb{N}$ such that $\{q^{j}(a+id):0\leq i,j\leq k\}\subset A$ and $\{b(c+ie)^{j}:0\leq i,j\leq k\}\subset A$ In this presentation we to extend these results for integral domains $\mathbb{Z}[i]$.

Stefanie Hittmeyer Bifurcations of generalised Julia sets near the complex quadratic family We consider a two-dimensional noninvertible map that was introduced by Bamon, Kiwi and Rivera in 2006 as a model of wild Lorenz-like chaos [1,2]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimage. The bounding critical circle and its images, together with the critical point and its preimages, form the critical set. In a specific parameter regime the map is a nonanalytic perturbation of the complex quadratic family. As parameters are varied away from the complex quadratic family the dynamics on the plane initially stay qualitatively the same. On the other hand, saddle points and their stable and unstable sets then appear as new ingredients of the dynamics. The stable, unstable and critical sets interact with the generalised Julia set, leading to the (dis)appearance of chaotic attractors and to dramatic changes in the topology of the generalised Julia set [3]. In particular, we find generalised Julia sets in the form of Cantor bouquets, Cantor tangles and Cantor cheeses. Continuation of these bifurcations in two parameters reveals a self-similar bifurcation structure near the period-doubling route to chaos of the complex quadratic family. [1] R. Bamon, J. Kiwi, and J. Rivera, Wild Lorenz-like attractors, Preprint, 2006, arXiv:math0508045v2 [2] S. Hittmeyer, B. Krauskopf and H.M. Osinga, Interacting global invariant sets in a planar map model of wild chaos, SIAM J. .Appl. Dyn. Syst., 12(3), 2013 [3] S. Hittmeyer, B. Krauskopf and H.M. Osinga, Interactions of the Julia set with critical and (un)stable sets in an angle-doubling map on C\{0}, Preprint, 2013

Dong Han Kim Subword complexity and Sturmian colorings of regular trees In this talk, we introduce subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and then introduce Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ``infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones. We will provide several examples. This is joint work with Seonhee Lim.

Bing Li Intermediate beta-shifts of finite type An aim of this talk is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\beta,\alpha} \colon x \mapsto \beta x + \alpha \bmod 1$ for which the corresponding intermediate $\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\beta,\alpha)$ such that the intermediate $\beta$-shift associated with $T_{\beta, \alpha}$ is a subshift of finite type and where the map $T_{\beta,\alpha}$ is not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\beta$-shifts and $\beta$-transformations, in that these two properties do not hold.

Yi-Chiuan Chen TBA Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.

Elisabete Alberdi Celaya Object Oriented Programming methodology for Dynamical Finite Element Method Computations E. ALBERDI CELAYA and J. J. ANZA Numerous phenomena of science and engineering are modelled mathematically using systems of Partial Derivative Equations (PDEs). The analytical solution of PDEs in a general domain is not possible and it is necessary to use numerical methods, among which the Finite Difference Method, the Boundary Element Method and the Finite Element Method (FEM) stand out, being the FEM the most capable in general, to deal with any shape domains and non linear problems. Traditionally, the development of numerical software has been based on the use of procedural languages, but in the last years there is an increasing interest of applying the paradigms of the Object Oriented Programming (OOP), which allows an efficient reutilization, extension and maintenance of the codes. After a short introduction to the mathematical modelling of the continuous media, which shows the similarity between the governing Partial Differential Equations in different applications, common blocks for Finite Element approximation are identified, and an Object Oriented Programming (OOP) methodology for linear and non linear, stationary and dynamic problems is presented. Advantages of this approach are commented and some results are shown as examples of this methodology. References: 1.- . Belytschko, W. K, Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons, Chichester, 2000. 2.- J. Bonet, R. D. Wood, Non linear continuum mechanics for finite element analysis, Cambridge University Press, Cambridge, 1997. 3.- K. F. Graff, Wave motion in elastic solids, Oxford University Press, London, 1975. 4.- E. Hairer, G. Wanner, Solving ordinary differential equations, II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996. 5.- T. J. R. Hughes, The Finite Element Method. Linear static and dynamic analysis, Prentice Hall, New Jersey, 1987. 6.- H. P. Langtangen, Computational Partial Differential Equations- Numerical Methods and Diffpack Programming, Springer-Verlag, Berlin, 1999. 7.- J. C. Simo, T. J. R. Hugues, Computational Inelasticity, Springer Verlag, New York, 1997.

Hoang Duc Luu Local metric entropy for finite time nonautonomous dynamical systems We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesin's entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special dynamical behavior such as stable and unstable manifolds and Lagrangian coherent structures.

Wenxiang Sun Degeneracy of entropy of flow with singurarity While entropy measures dynamical complexity and an invariant for disrcete systems, entropy is degenerate for flows with singurarity: there exists an equivalent pair of flows with singularity such that one has zero entropy and the other has infinite entropy. Such entropy degeneracy happens if and only if all invariant ergodic measures with positive entropy near singularities loss their invarance, when shiftting equivalently one flow to another.

Sergey Tikhomirov Shadowing in actions of non-abelian groups We introduce notion of shadowing property for actions of finitely generated not necessarily abelian groups. In contract with shadowing for diffeomorphisms and flows we show that shadowing property depends not only on hyperbolicity but on the group structure as well. For nilpotent groups we prove analog of shadowing lemma. We give an example of an action of a solvable group, whose shadowing property depends on quantitative properties of hyperbolicity. Finally we prove that there is no linear action of free nonabelian group which has shadowing property.

Sergei Pilyugin Shadowing and structural stability of diffeomorphisms and flows We discuss several recent results on relations between shadowing and structural stability of diffeomorphisms and flows.

Alba Marina Málaga Sabogal Generic minimality and almost sure conservativity for a family of dynamical systems on the cylinder S¹×Z I work on a family of (discrete) dynamical systems which is heuristically related to a billiard on a parallelogram. This family is defined on the discrete cylinder $\mathbb S^1\times \mathbb Z$ where $\mathbb T^1={\mathbb R/\mathbb Z}$ is the one-dimensional torus (i.e. the circle). For any bi-infinite sequence $\underline\alpha\in\mathbb T^\mathbb Z$, we define the transformation $F_{\underline\alpha}$ almost everywhere on the cylinder as follows:$$F_{\underline\alpha}\left([x]_\mathbb Z,n\right)=\left([x+\alpha_n]_\mathbb Z, n+\left\{\begin{array}{rcl}1&if &x+\alpha_n\in(0,\frac12)+\mathbb Z\\-1&if&x+\alpha_n\in(-\frac12,0)+\mathbb Z\\ \end{array}\right.\right).$$When the sequence $\underline\alpha$ is constant and irrational, Conze and Keane showed in {[1]} that $F_{\underline\alpha}$ is ergodic.\\I am trying to understand what are the typical properties of $F_{\underline\alpha}$ in the following meaning. Namely, what properties hold for almost any $\underline\alpha$ or for a generic $\underline\alpha$ in the parameter space? For the moment I have proved that conservativity is both generic and almost-sure, whereas the minimality is generic. I would like to understand also the diffusion properties of this family. Bibliography Conze, J.-P.; Keane, M. Ergodicité d'un flot cylindrique. (French) Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976), Exp. No. 5, 7 pp. Dépt. Math. Informat., Univ. Rennes, Rennes, 1976.

Elisabete Alberdi Celaya Application of some numerical damping control multistep methods E. ALBERDI CELAYA and J.J. ANZA AGUIRREZABALA The second order Ordinary Differential Equation (ODE) system obtained after semidiscretizing the wave-type partial differential equation (PDE) with the finite element method (FEM) shows strong numerical stiffness. Its resolution requires the use of numerical methods with good stability properties and controlled numerical dissipation in the high-frequency range. Some of the methods developed in the linear range are the Collocation method, the Wilson method, the HHT-$alpha$ method, the Houbolt method, or more recent works as the generalized-alfa method. The development of similar methods for nonlinear problems is more recent and it has been originated by the presence of numerical instabilities when solving nonlinear problems by methods which are unconditionally stable in the linear range. In this paper, we work in the linear range and we formulate a new method called BDF-$alpha$, which is second-order accurate, unconditionally stable for some values of the parameter $alpha$ and permits a parametric control of numerical dissipation. Some applications of this new method are shown. References: 1.- F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for nonlinear dynamics. Part I: low order methods for two model problems and nonlinear elastodynamics, Comput. Meth. Appl. Mech. Eng., Vol.190, 2603-2649, 2000. 2.- F. Armero, I. Romero, On the formulation of high-frequency dissipative time stepping algorithms for nonlinear dynamics. Part II: second order methods, Comput. Meth. Appl. Mech. Eng., Vol.190, 6783-6824, 2001. 3.- J. Chung, G. M. Hulbert, A time integration algorithm for structural dynamics with improvednumerical dissipation: the generalized-$alpha$ method, Journal of Applied Mechanics, Vol.60, 371-375, 1993. 4.- O. Gonzalez, Exact energy-momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Eng., Vol.190, 1763-1783, 2000. 5.- E. Hairer, G. Wanner, Solving ordinary differential equations, II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996. 6.- H. M. Hilber, T. J. R. Hughes, Robert L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn., Vol.5, 283-292, 1977. 7.- T. J. R. Hughes, The finite element method. Linear Static and dynamic finite element analysis, Prentice-Hall International Editions, New Jersey, 1987.

Piotr Oprocha Proximal and completely scrambled systems A dynamical system $(X,f)$ is completely scrambled if every pair $x,y$ of distinct points in $X$ is proximal (i.e. $\liminf d(f^n(x),f^n(y))=0$) but not asymptotic (i.e. $\limsup d(f^n(x),f^n(y))>0$). In 1999, Huang and Ye proved that continua of arbitrary dimension admits completely scrambled homeomorphism. In this talk we aim to present a brief characterization of completely scrambled homeomorphisms, ideas behind the construction of Huang and Ye and some other methods of construction of completely scrambled systems which are in fact motivated by much earlier papers (on a different topic). If time permits, we will present a method to answer a question left open in the Huang and Ye's paper (joint work with Fory\'s, Huang and Li). Finally, we will present a few questions for further research.

Christian Rodrigues TBA Amongst the main concerns of Dynamics one wants to decide whether asymptotic states are robust under random perturbations. Considering the iteration of $f : M \to M$, such randomness is implemented by a family $\{p_{\varepsilon}(\, \cdot \,|x)\}$ of Borel probability measures, such that every $p_{\varepsilon}(\, \cdot \,|x)$ is supported inside the $\varepsilon$-neighbourhood of $f(x)$. Alternatively, the orbit is given by the iteration $x_{j} = g_{j} \circ \cdots \circ g_{1}(x_{0})$, where each measurable $g_{j}$ is picked at random $\varepsilon$-close from the original map $f$. Endowing the collection of maps $\{g_{j}\}$ with a probability distribution $\nu_{\varepsilon}$, we say that the sequence of random maps is a representation of that Markov chain if for every Borel subset $U$, $p_{\varepsilon}(U|x) = \nu_{\varepsilon}(\{g : g(x) \in U \})$. In this talk we systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. From this scheme, we not only deduce the representation by measurable and continuous random maps, but also obtain conditions for the to construct random diffeomorphisms from a given Markov chain. This is a joint work with Jost, and Kell.

Dawei Yang Typical dynamics of three-dimensional I will talk about what are the typical dynamics of the flows generated by three-dimensional vector fields. And I will discuss how to get singular hyperbolicity from domination.

Peter Kloeden Semi-Hyperbolicity and Bishadowing Semi-hyperbolicity extends the classical concept of hyperbolicity for diffeomorphisms to noninvertible Lipschitz mappings on sets that need not be invariant. Moreover, the stable and unstable manifolds need to be mapped only approximately onto their counterparts. Semi-hyperbolic mappings are bi-shadowing, i.e., they also satisfy a converse form of shadowing in which the pseudo-orbits are not arbitrary but are the true orbits of admissible classes of approximating mappings. The theory of these concepts is given and illustrated through applications of nonsmooth systems including those with delays and hysteresis, as well as numerical approximations.

In-Jee Jeong Outer Billiards with Contraction We consider the outer billiards map around a convex polygon, but now composed with a linear contraction by $lambda$. Experiments suggest that for almost every polygons and values of $lambda$, there are only finitely many periodic orbits to which all orbits are attracted. We prove these "regularity" results in the case when the polygon is either a triangle, parallelogram, or a regular hexagon. On the other hand, it turns out that for every quadrilateral, there is a special value of $lambda$ where the dynamics is reduced to that of a piecewise contractive map of the interval. From this, we can show that for certain quadrilaterals there is an attracting Cantor set. The ergodic properties will be discussed as well.

Nandadulal Bairagi Nonlinear dynamics in relation to mathematical ecology and epidemiology TBA

Valery Gaiko Global limit cycle bifurcations in polynomial dynamical systems The global qualitative analysis of polynomial dynamical systems is carried out. Using a new bifurcational geometric method, we solve the problem of the maximum number of limit cycles surrounding a singular point for an arbitrary planar polynomial system and Hilbert's Sixteenth Problem for the general Lienard polynomial system with an arbitrary (but finite) number of singular points. Applying this method, we study also three-dimensional polynomial systems and, in particular, complete the strange attractor bifurcation scenario in the classical Lorenz system connecting globally the homoclinic, period-doubling, Andronov-Shilnikov, and period-halving bifurcations of its limit cycles.

Dominik Kwietniak Specification-like properties and invariant measures I would like to present a survey of various properties similar to Bowen's specification property. In a joint work with Katrin Gelfert we proposed yet another generalization, which is strong enough to imply that the set of ergodic measures is dense in the simplex of all invariant measures and every invariant measure has a generic point. I will describe our approach and provide examples which allow to distinguish between our approach and old and recent specification-like methods of Sigmund, Bowen, Climenhaga-Thompson, Pfister-Sullivan.

Edson Vargas Invariant measures for Cherry flows We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the casewhen the saddle is dissipative or conservative we showthat the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle. Joint work with Radu Saghin.

CA Morales Sectional-hyperbolic systems We review progress in the theory of sectional-hyperbolic dynamics.

Rafael POTRIE Quasi-attractors for 3-dimensional diffeomorphisms far from tangencies We discuss finiteness and uniqueness results of quasi-attractors in several contexts for diffeomorphisms of 3-manifolds which are far from homoclinic tangencies. This builds on colaborations with S. Crovisier, A. Hammerlindl, M. Sambarino.

Shin Kiriki Blenders in center unstable H'enon-like families: with an application to heterodimensional bifurcations We give an explicit family of polynomial maps called center unstable H'enon-like maps and prove that they exhibits blenders for some parameter values. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle. This is a joint work with L. J. Díaz and K. Shinohara, to appear in Nonlinearity.

Khosro Tajbakhsh Asymptotic Measure Expansive Diffeomorphisms In this brief paper the notion of asymptotic measure expansiveness and its relationship with the dominated splitting is considered. It is proved that if a diffeomorphism $f$ admits a dominated splitting then it is asymptotic measure expansive. Also, a diffeomorphism with a homoclinic tangency perturbed is reaching to a non-asymptotic measure expansive diffeomorphism.