VCDS-8 Budapest, June 24-28, 2019
8th Visegrad Conference on Dynamical Systems
Programme
This event is sponsored by the Budapest University of Technology and Economics, ELTE EotvosLorand University, the Hungarian Academy of Sciences, the Renyi Institute and the VisegradFund.
VCDS-8 Participants 1
List of Participants
Anusić, Ana, Brazil, University of Sao PauloBalibrea, Francisco, Spain, Facultad de Matematicas en Campus de EspinardoSewell, Benedict, United Kingdom, University of WarwickBerkane, Ali, Algeria, High school of Management sciencesBobok, Jozef, Czech Republic, Czech Technical University in PragueBose, Debashish, India, Shiv Nadar UniversityBouchelaghem, Faycal , Algeria, High School of Management SciencesBruin, Henk, Austria, University of ViennaBuczolich, Zoltan, Hungary, ELTE Eotvos Lorand UniversityCinc, Jernej, Poland, AGH Krakow and IT4Innovations University of OstravaDıaz, Mauricio , Chile, Universidad del Bio BioDrwięga, Tomasz, Poland, AGH University of Science and TechnologyDymek, Aurelia, Poland, Nicolaus Copernicus UniversityEisner, Tanja, Germany, University of LeipzigFarkas, Abel, Hungary, Renyi InstituteFuhrmann, Gabriel, United Kingdom, Imperial College LondonGuzik, Grzegorz, Poland, AGH University of Science and TechnologyPourmand, Habibeh, Poland, Jagiellonian UniversityHauser, Till, Germany, Friedrich-Schiller-Universitat JenaHric, Roman, Slovakia, Matej Bel UniversityHua, Yongxia, China, Southern University of Science and TechnologyJimenez Lopez, Victor, Spain, Universidad de MurciaKeszthelyi, Gabi, Hungary, RenyiKocan, Zdenek, Czech Republic, Silesian University in OpavaKovac, Jozef, Slovakia, Comenius University in BratislavaKunszenti-Kovacs, David, Hungary, MTA Renyi Alfred Matematikai IntezetKupka, Jiri, Czech Republic, University of OstravaKwietniak, Dominik, Poland, Jagiellonian University in KrakówŁącka, Martha, Poland, Jagiellonian University in KrakowLIAN, ZHENGXING, Poland, Instytut Matematyczny Polskiej Akademii NaukLiang, Chao, China, Central University of Finance and EconomicsLukina, Olga, Austria, University of ViennaMakhrova, Elena, Russia, Lobachevsky State University of Nizhni NovgorodMiller, Alica, United States of America, University of LouisvilleMisiurewicz, Michał, United States of America, Indiana University-Purdue University IndianapolisMoore, Ryo, Chile, Pontificia Universidad de Catolica de ChilePravec, Vojtech, Czech Republic, Silesian University in OpavaProkaj, Rudolf Daniel, Hungary, Budapest University of Technology and EconomicsRaith, Peter, Austria, Universitat WienRams, Michal, Poland, Mathematical Institute of the Polish Academy of SciencesSell, Daniel, Germany, Friedrich-Schiller-Universitat JenaSimon, Karoly, Hungary, Budapest University of Technology and EconomicsSimonne Bakos, Erzsebet, Hungary, Kerek utcai altalanos iskola es gimnaziumSlawinska, Joanna, United States of America, Department of Physics, University of Wisconsin-MilwaukeeSnoha, Lubomır, Slovakia, Matej Bel University
VCDS-8 Participants 2
Śpiewak, Adam, Poland, University of WarsawSun, Peng, China, Central University of Finance and EconomicsSzasz , Domokos, Hungary, Budapest University of TechnologySzekely, Zoltan, Guam, University of GuamTesarcık, Jan, Czech Republic, Silesian Univerzity in OpavaTetenov, Andrei, Russia, Novosibirsk state university Gorno-Altaisk state universityTroscheit, Sascha, Austria, University of ViennaVejnar, Benjamin, Czech Republic, Charles UniversityVertesy, Gaspar, Hungary, Eotvos Lorand UniversityWeiss, Benjamin, Israel, Hebrew University of JerusalemSun, Wenxiang, China, Peking UniversityWolf, Christian, United States of America, The City College of New York
(This list and talk/poster abstratcts are ordered according to the simplified last names providedat the registration. If someone provided his first name as last name then it was used in the ordering.)
VCDS-8 Schedule 3
Monday, June 24
08:00 - 10:00 Registration08:55 - 09:05 Opening of the conferenceSession Chair: Lubomır Snoha09:05 - 09:55 Benjamin Weiss Bernoulli Disjointness10:05 - 10:25 Francisco Balibrea On positions of periodic points on plane dynamical systems10:30 - 11:00 Coffee break11:00 - 11:50 Henk Bruin Li-Yorke chaos from the Lebesgue measure point of view.12:00 - 12:20 Ana Anusić Inhomogeneities in unimodal inverse limit spaces12:25 - 14:00 LunchSession Chair: Peter Raith14:00 - 14:20 Jernej Cinc On the Barge entropy conjecture14:25 - 14:45 Gabriel Fuhrmann The structure of mean equicontinuous group actions14:50 - 15:10 Olga Lukina Continuous orbit equivalence for actions of nilpotent groups15:15 - 15:45 Coffee breakSession Chair: Andrei Tetenov15:45 - 16:05 Zoltan Szekely Permutation Doubling in One-Dimensional Dynamical Systems16:10 - 16:30 Joanna Slawinska Vector-Valued Spectral Analysis of Complex Flows16:35 - 16:55 Wenxiang Sun Entropy Black Hole17:00 - 17:20 Mauricio Dıaz Relation between Sensitive system and MDS using Furstenbergfamily
Tuesday, June 25
Session Chair: Henk Bruin09:05 - 09:55 Jozef Bobok Constant slope, entropy and horseshoes for a map on a tame graph10:05 - 10:25 Andrei Tetenov On topological classification of PCF self-similar dendrites10:30 - 11:00 Coffee break11:00 - 11:50 Lubomır Snoha On the problem of characterizing minimal spaces/sets12:00 - 12:20 Aurelia Dymek Generalization of B-free Systems12:25 - 14:00 LunchSession Chair: Roman Hric14:00 - 14:20 Tomasz Drwięga On chaos on dendrite14:25 - 14:45 Elena Makhrova The structure of dendrites and dynamics of continuous maps onthem14:50 - 15:10 Zdenek Kocan On properties of dynamical systems on graphs and dendrites15:15 - 15:45 Coffee breakSession Chair: Christian Wolf15:45 - 16:05 Alica Miller Recent results about sensitivity in general semiflows16:10 - 16:30 Benjamin Vejnar Complexity of dynamical systems16:35 - 16:55 Peng Sun Entropy and gluing orbit property17:00 - 17:20 Poster session
VCDS-8 Schedule 4
Wednesday, June 26
Session Chair: Jozef Bobok09:05 - 09:55 Domokos Szasz Heat Conduction from Hamiltonian Dynamics10:05 - 10:25 Christian Wolf Geometry and computability at zero temperature10:30 - 11:00 Coffee break11:00 - 21:00 Excursion to Visegrad
Thursday, June 27
Session Chair: Michal Rams09:05 - 09:55 Tanja Eisner Weighted Ergodic Theorems10:05 - 10:25 Peter Raith On mixing properties for a class of expanding Lorenz maps10:30 - 11:00 Coffee break11:00 - 11:50 Dominik Kwietniak On Problem 32 from Rufus Bowen’s list: classification of shiftspaces with specification12:00 - 12:20 Jiri Kupka Differentiable periodic point free minimal dynamical systems withvanishing derivative everywhere12:25 - 14:00 LunchSession Chair: Zdenek Kocan14:00 - 14:20 Ryo Moore Properties of Birkhoff spectra for the generic continuous functions ona shift space14:25 - 14:45 Sascha Troscheit The Assouad spectrum and fine information on extremal scaling14:50 - 15:10 Roman Hric Minimality for actions of abelian (semi)groups15:15 - 15:45 Coffee breakSession Chair: Jiri Kupka15:45 - 16:05 Grzegorz Guzik On a class of operators on the product of spaces of measureshaving the unique invariant distribution (joint work with Rafal Kapica)16:10 - 16:30 Jozef Kovac Distributional chaos on the space of probability measures16:35 - 16:55 Chao Liang Openness of non-uniformly hyperbolic diffeomorphisms19:00 - 22:00 Conference dinner
Friday, June 28
Session Chair: Tanja Eisner09:05 - 09:55 Michal Rams Mass transference principle for general shapes10:05 - 10:25 Victor Jimenez Lopez Global asymptotic stability of a generalization of the Pieloudifference equation10:30 - 11:00 Coffee break11:00 - 11:50 Michał Misiurewicz Renormalization towers and their forcing12:00 - 12:20 Jan Tesarcık On Spectrum of Piecewise monotonic maps with Markov condition12:25 - 14:00 LunchSession Chair: Dominik Kwietniak14:00 - 14:20 Till Hauser Some comments on the Entropy of Delone sets14:25 - 14:45 Vojtech Pravec Remarks on definitions of periodic points for nonautonomousdynamical system14:50 - 15:10 Daniel Sell Uniformity of cocycles for subshifts with leading sequences15:15 - 15:45 Coffee break
VCDS-8 Talks 5
Inhomogeneities in unimodal inverse limit spaces
Ana Anusić
University of Sao Paulo
We will discuss how the dynamics of the bonding map affects the structure of inhomogeneities(points locally not homeomorphic to Cantor set of arcs) in unimodal inverse limit spaces. Speci-fically, we will address the problem of characterizing unimodal inverse limits for which the set ofinhomogeneities corresponds to the set of endpoints. It turns out that the crucial notion in thatcharacterization is persistent recurrence of the critical point, introduced by Blokh and Lyubich in1991. We will also discuss possible directions in generalizing this notions and results to generalone-dimensional continua.This is a joint work with Lori Alvin, Henk Bruin, and Jernej Cinc.
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On positions of periodic points on plane dynamical systems
Francisco Balibrea
Facultad de Matematicas en Campus de Espinardo
We will revise some examples of plane dynamical systems such as those given by the equations
La(x, y) = (y, ay(1− x))
Fa,b(x, y) = (y, ay − b− x2)T (x, y) = (x(4− x− y), xy)
which have been considered in the literature since a long term.
In such examples, particularly in those depending of real parameters, usually new periodic pointsappear when the parameters grow and belong to some invariant subsets of R2 which in turn forma periodic set of them. Usualy such appearances are associate to Hopf bifurcations.
The alternative appearances and disapearances of invariant curves in the above bifurcation andat the same time, the change of stability of the periodic points, are the common mechanisms forunderstand their positions.
We will do a brief explanation of such facts in the examples and present some pictures of them.
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VCDS-8 Talks 6
Constant slope, entropy and horseshoes for a map on a tame graph
Jozef Bobok
Czech Technical University in Prague
We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a specialPeano continuum for which the set containing branchpoints and endpoints has countable closure.In our investigation we confine ourselves to the countable Markov case. We show a necessary andsufficient condition under which a locally eventually onto, countably Markov map f of a tamegraph G is conjugate to a map g of constant slope. In particular, we show that in the case ofa Markov map f that corresponds to a recurrent transition matrix, the condition is satisfied forconstant slope ehtop(f), where htop(f) is the topological entropy of f . Moreover, we show that inour class the topological entropy htop(f) is achievable through horseshoes of the map f .[1] A. Bartos, J. Bobok, P. Pyrih, S. Roth, B. Vejnar, Ergodic Theory and Dynamical systems,published online http://dx.doi.org/10.1017/etds.2019.29
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Li-Yorke chaos from the Lebesgue measure point of view.
Henk Bruin
University of Vienna
Given a dynamical system T : X → X on a compact metric space (X, d), a Li-Yorke pair is a pairof points x, y ∈ X such that
lim infn→∞ d(T
n(x), T n(y)) = 0 and lim supn→∞
d(T n(x), T n(y)) > 0.
If there is an uncountable set S in which all two distinct point form a Li-Yorke pair, then T iscalled Li-Yorke chaotic.In this talk I want to study Li-Yorke chaos in terms of Lebesgue measure prevalence of Li-Yorkepairs (joint with Jimenez-López [1]) and Li-Yorke tuples (joint with Oprocha [2]).
[1] H. Bruin, V. Jimenez-López, On the Lebesgue measure of Li-Yorke pairs for interval maps,Communications in Mathematical Physics, 299 (2010), 523-560.
[2] P. Oprocha, On ”observable” Li-Yorke tuples for interval maps, Nonlinearity, 28 (2015), 1675-1694.
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VCDS-8 Talks 7
On the Barge entropy conjecture
Jernej Cinc
AGH Krakow and IT4Innovations University of Ostrava
In 1920’s Knaster described the first example of a hereditarily indecomposable continuum whichlater received the name pseudo-arc due to the property that every of its subcontinua is homeomor-phic to itself, yet the continuum is not homeomorphic to the arc. One of the basic questions intopological dynamics about such complicated spaces is, what are the possible topological entropiesof their homeomorphisms. For the pseudo-arc the conjecture that possible topological entropiesof its homeomorphisms can take any positive real value was posed by Marcy Barge in 1989. Untilnow, all known pseudo-arc homeomorphisms have had entropy 0 or∞. In this talk, I will overviewsome known results relating to the Barge entropy conjecture and discuss our recently obtainedpositive solution to the conjecture. The positive entropy homeomorphisms that we construct areperiodic point free, except for a unique fixed point.The talk is based on a joint work with Jan P. Boroński and Piotr Oprocha.
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VCDS-8 Talks 8
Relation between Sensitive system and MDS using Furstenberg family
Mauricio Dıaz
Universidad del Bio Bio
In this article we going to use a Measurable dynamical system denoted as (X,B,µ,T,S) for give ananalysis about a relation between the iterated discrete dynamics that are present in continuousself maps on X and a strong relation between Systems with Sensitive properties and the collectionof additive actions generated by a hitting time set and other measurable factors as the topologicalentropy and additional behaviours in ω-limit. Using Furstenberg family we going to prove thata F-scattering and F-recurrent point has a dense Gδ subset of X × X/(∆ 1
δ′) for some δ′ > 0 for
MDS for Baire sets with second category. Finally we going to prove if S ∼= Z, then MDS has localchaos for every cyclic group with dual sets of F ′ip ∩ F ′ps ∩ F ′pd1, being F’ an equivalent family of Fgenerated by upper Banach density 1.
[1] Li, J. Transitive points via Furstenberg family, Topology and its Applications, 158 (16) (2011),2221-2231.
[2] Huang. W, Ye. X. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos, Topology and itsApplications , 117(3) , (2002).259-272.
[3] Li, J., Li, J. and Tu, S. Devaney chaos plus shadowing implies distributional chaos, Chaos: AnInterdisciplinary Journal of Nonlinear Science, 26(9) , (2016).93-103
[4] Shao, S. Proximity and distality via Furstenberg families, Topology and its Applications ,153(12) , (2006).2055-2072.
[5] Huang, W., Khilko, D., Kolyada, S., & Zhang, G, Dynamical compactness and sensitivity,Journal of Differential Equations 260, (2016).
[6] Li, J., & Tu, S., On proximality with Banach density one. ,Journal of Mathematical Analysisand Applications 416(1), (2014). 36-51
[7] Pawlak, R. J., & Loranty, A. , On the local aspects of distributional chaos. , Chaos: An Inter-disciplinary Journal of Nonlinear Science, 29(1), (2019). 013-104
[8] Blanchard, F., Glasner, E., Kolyada, S., & Maass, A, On Li-Yorke pairs ,Journal fur die reineund angewandte Mathematik 547, (2002). 51-68
[9] Diaz, M., Relation between sensitive systems, topological entropy and Baire set in MDS. , ,(2018).
[10] Downarowicz, T., Positive topological entropy implies chaos DC2. Proceedings of the AmericanMathematical Society, 142(1) , (2014).137-149.
[11] Vasisht, R., & Das, R. A note on F-sensitivity for non-autonomous systems. Journal ofDifference Equations and Applications, , (2019).1-12.
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VCDS-8 Talks 9
On chaos on dendrite
Tomasz Drwięga
AGH University of Science and Technology
In this talk I will present questions left open in [Z. Kocan, Internat. J. Bifur. Chaos Appl. Sci.Engrg. 22, article id: 125025 (2012)] for dendrite maps:
• whether there is a relation between ω-chaos and distributional chaos,
• whether there is a relation between an infinite LY-scrambled set and distributional chaos,
• whether there is a relation between an infinite LY-scrambled set and ω-chaos,
together with their answers. This provides complete characterization of chaos on dendrites. Thetalk is partially based on joint work with Piotr Oprocha.
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Generalization of B-free Systems
Aurelia Dymek
Nicolaus Copernicus University
Let B ⊂ 2, 3, . . .. The set of multiples is defined as
MB =⋃
b∈BbZ.
Such sets were studied in the 1930s by Behrend, Bessel-Hagen, Chowla, Erdos and others. Theset of so called B-free numbers is FB = Z \ MB. We denote by η ∈ 0, 1Z the characteristicfunction of FB and consider the orbit closure Xη of η in the space 0, 1Z with the left shift S.The subshift (Xη, S) is called B-free system. Recently, B-free systems are under intensive study.In the joint paper with Kasjan, Kułaga-Przymus and Lemańczyk we showed that the system(Xη, S) is proximal if and only if B contains an infinite subset of pairwise coprime integers. Somegeneralizations of B-free systems were studied by Cellarosi, Vinogradov, Baake and Huck. One ofthem is to replace Z by Zd, d 2 and B by a family of lattices (subgroups with finite index) in Zdand the left shift S by the multidimensional shift (Sn)n∈Zd . To show some difficulties to generalizeone-dimensional theorems, I will discuss the problem of proximality in the multidimensional case.In particular, I will show that the characterization of proximality can be obtained in the casewhen the multiplicative structure is present. I will formulate some open questions related tomultidimensional B-free systems.
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VCDS-8 Talks 10
Weighted Ergodic Theorems
Tanja Eisner
University of Leipzig
We present an overview on good weights for the pointwise ergodic theorem.
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The structure of mean equicontinuous group actions
Gabriel Fuhrmann
Imperial College London
This talk deals with mean equicontinuous actions of locally compact σ-compact amenable groupson compact metric spaces.After a motivation, introduction and brief review of the concept of mean equicontinuity, we willdiscuss alternative characterisations. These include regularity properties of the projection onto themaximal equicontinuous factor, a description of the associated product action as well as spectralproperties. As one application of our results, we obtain an interesting class of non-abelian meanequicontinuous examples.This is a joint work with Maik Groger and Daniel Lenz.
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On a class of operators on the product of spaces of measures having the uniqueinvariant distribution (joint work with Rafal Kapica)
Grzegorz Guzik
AGH University of Science and Technology
Let m be fixed positive integer and X be a Polish space. We consider cotractive (in some sense)mappings acting on the product of m copies of the space M(X ) of all finite Borel measures onX withe values in M(X ). We show that such operators admit a unique generalized fixed pointwhich is a probability measure with the first moment finite. Moreover, such a measure attractsall ditributions in the Hutchinson-Wasserstein norm and the rate of convergence is geometric.Obtained results we apply to generalized iterated function systems (GIFSs) as well as to stochasticdifference equations with multiple delays.
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VCDS-8 Talks 11
Some comments on the Entropy of Delone sets
Till Hauser
Friedrich-Schiller-Universitat Jena
In order to study Delone sets of finite and infinite local complexity in Euclidean space, we presentthat a version of the Ornstein Weiss Lemma holds for any compactly generated locally compactabelian group. From this, we conclude that patch counting entropy can be calculated as a limitalong certain Van Hove sequences.
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Minimality for actions of abelian (semi)groups
Roman Hric
Matej Bel University
I will present selected problems and results from a joint work with L’. Snoha: Dense orbits indiscrete and continuous systems on topological spaces (Preprint 2018), and a joint work with M.Dirbak, P. Malicky, L’. Snoha and V. Spitalsky: Minimality for actions of abelian semigroups oncompact spaces with a free interval, Ergodic Theory & Dynamical Systems (First published online2018).In the first work, we focus on topological transitivity and minimality of maps, homeomorphismsand (semi)flows on topological spaces—in general, noncompact ones. A special emphasis is givenon hereditariness of density of orbits from (semi)flows to time-t maps.In the second work, we investigate minimality of actions of general abelian semigroups on compactspaces possesing a free interval, i.e. an open subset of the space homeomorphic to the open interval.We characterise such spaces admitting a minimal action of a given abelian semigroup and we alsodescribe the topological structure of minimal sets intersecting a free interval.
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VCDS-8 Talks 12
Global asymptotic stability of a generalization of the Pielou difference equation
Victor Jimenez Lopez
Universidad de Murcia
In this talk (which is an account of some of the results in [1]) we deal with the second-orderdifference equation
xn+1 =(α + 1)xγnα + xn−1
, n = 0, 1, 2, . . . ,
where α, γ are positive real numbers and we consider positive initial conditions x0, x−1. It generali-zes, modulo a linear change of variables, the well-known Pielou equation (corresponding to γ = 1)and can be seen as a population model where the per capita growth rate xn+1/xn is the productof two density-dependent factors: the delayed factor of intraspecific competition (α + 1)/(α + x)already present in the classical Pielou equation, and a new factor xγ−1, which can be a cooperationfactor (γ > 1) or an interference factor (0 < γ < 1).Our precise aim is to investigate the global asymptotical stability of the unique equilibrium p = 1of the equation, that is, whether p may be stable and simultaneously attract all the solutions ofthe equation. It is easily seen that if γ > 1, then the cooperation factor induces extinction belowa certain threshold (the so-called Allee effect), hence p cannot be globally attracting. Our mainresult:
Theorem. If γ ¬ 1, then the equilibrium p = 1 is globally asymptotically stable.In this way we generalize the analogous result for the Pielou equation (γ = 1) earlier proved in [2].Our proof is based on a rather natural combination of two techniques which could be, in principle,applicable to other global asymptoticity problems: if α is not too small, then we “dominate” thedynamics by that of a first-order equation; if α is very small, then we find a first integral for thelimit equation α = 0, which turns out to be an “almost” Lyapunov function for our equation.
[1] A. Garab, V. Jimenez López and E. Liz, Global asymptotic stability of a generalization of thePielou difference equation, Mediterr. J. Math. (to appear).
[2] S. A. Kuruklis and G. Ladas, Oscillations and global attractivity in a discrete delay logisticmodel, Quart. Appl. Math. 50 (1992), 227–233.
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VCDS-8 Talks 13
On properties of dynamical systems on graphs and dendrites
Zdenek Kocan
Silesian University in Opava
We consider a list of properties of continuous maps on compact metric spaces, e.g. the existenceof an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory,the existence of an omega-limit set containing two minimal sets, and others. Here we present theknown relations between the properties in the case of graph maps and in the case of dendritemaps.It is based on a joint work with Veronika Kurkova and Michal Malek.
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Distributional chaos on the space of probability measures
Jozef Kovac
Comenius University in Bratislava
Let M be the space of all probability measures on the interval [0, 1] endowed with the borelσ−algebra and let T ⋆ :M→M be a map induced by the random dynamical system
xn+1 =
f(xn) with probability p,
g(xn) with probability 1− p,
where f, g : [0, 1] → [0, 1] are continuous maps and p ∈ (0, 1) (for a measurable set C,T ⋆µ(C) ≡ pµ(f−1(C)) + (1 − p)µ(g−1(C))). We will focus on the relationship between distribu-tional chaos in the random dynamical system (as defined in [1]) and the distributional chaos inthe system (M, T ⋆).
[1] J. Kovac, K. Jankova. Distributional chaos in random dynamical systems, Journal of
Difference Equations and Applications, 25(4)(2019), 455-480.
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VCDS-8 Talks 14
Differentiable periodic point free minimal dynamical systems with vanishingderivative everywhere
Jiri Kupka
University of Ostrava
Let R denote the set of real numbers. K. Ciesielski and J. Jasinski in [1] recently proved that thereexist a non-empty compact perfect set X ⊆ R and a differentiable bijection f : X → X such thatf ′(x) = 0 for every x ∈ X. It was also shown that f is a minimal dynamical system. Our workwas motivated by this result.In this talk we discuss and answer the following question: Can every minimal Cantor dynamicalsystem be embedded into the real line with zero derivative everywhere? This presentation is, infact, a continuation of the talk presented during VCDS 2017 (based on results in [2]) in whichwe discussed an analogous question without the assumption of minimality. The talk is based on ajoint work with J. P. Boroński and P. Oprocha.
[1] K.C. Ciesielski and J. Jasinski, An auto-homeomorphism of a Cantor set with zero derivativeeverywhere, J. Math. Anal. Appl. 434(2) (2016), 1267–1280.
[2] J.P. Boroński, J. Kupka and P. Oprocha, Edrei’s Conjecture Revisited, Annales Henri Poincare19(1) (2018), 267–281.
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On Problem 32 from Rufus Bowen’s list: classification of shift spaces withspecification
Dominik Kwietniak
Jagiellonian University in Kraków
Rufus Bowen left a notebook containing 157 open problems and questions. Problem 32 on thatlist asks for classification of shift spaces with the specification property. Unfortunately, there is nouniversally accepted agreement what does it mean “to classify” a family of mathematical objects,and Bowen didn’t left any clues. During my talk, I will describe one of the most popular waysof making the problem formal. It is based on the theory of Borel equivalence relations. Insidethat framework, I will explain a result saying that (roughly speaking) there is no reasonableclassification for shift spaces with the specification property. More precisely, I will show thatthe isomorphism relation on the space of shifts with the specification property is a universalcountable Borel equivalence relation, i.e. for every countable Borel equivalence relation F, wehave that F is Borel reducible to E. It follows that no classification using a finite set of definableinvariants is possible. This solves the problem provided that Bowen would agree with the notionof “classification” provided by the theory of Borel equivalence relations.
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VCDS-8 Talks 15
Openness of non-uniformly hyperbolic diffeomorphisms
Chao Liang
Central University of Finance and Economics
In this talk I will introduce some results on the Cr-topological properties of the subset of nonu-niformly hyperbolic diffeomorphisms in a certain class of partially hyperbolic symplectic systems(or volume-preserving systems).
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Continuous orbit equivalence for actions of nilpotent groups
Olga Lukina
University of Vienna
The notion of continuous (topological) orbit equivalence for group actions on Cantor sets wasintroduced by Boyle in his thesis. Boyle also proved the first rigidity result in this area, namely,that two actions of the group of integers on a Cantor set C are continuously orbit equivalent ifand only if they are flip conjugate.More recently, continuous orbit equivalence for equicontinuous actions of finitely generated groupson a Cantor set C was studied by Li and by Cortez and Medynets. In both works, an importantassumption is that the actions are topologically free, which means that the set of points withtrivial stabilizers is dense in C.In this talk, we concentrate on actions of finitely generated groups on a Cantor set C whichare not topologically free. Such actions are ubiquitous in mathematics, arising, for example, asactions of iterated monodromy groups in geometric group theory, actions associated to arborealrepresentations in arithmetic dynamics, and in other contexts. Such actions can be divided intotwo classes, stable and wild. An action is stable if the induced holonomy action on a sufficientlysmall clopen subset of C is topologically free.In the talk, we concentrate on actions of nilpotent groups. We show that if a group G is nilpotentand finitely generated, then its action is always stable. This leads to a rigidity result, which statesthat if two actions by finitely generated nilpotent groups are continuously orbit equivalent, thenthey are return equivalent.Joint work with Steve Hurder.
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VCDS-8 Talks 16
The structure of dendrites and dynamics of continuous maps on them
Elena Makhrova
Lobachevsky State University of Nizhni Novgorod
By continuum we mean a compact connected metric space. Dendrite is a locally connected conti-nuum without subsets homeomorphic to a circle.Let X be a dendrite. The next properties are well known (see, e.g., [1] – [2]):1) a dendrite is one-dimensional continuum;2) for any points x, y ∈ X there is a unique arc containing these points;3) the set of branch points of X is at most countable;4) the number of connected components of the set X \ p is at most countable for any pointp ∈ X.In spite of last two properties there are dendrites with complicated structure. For example, Gehmandendrite has uncountable set of end points or there are dendrites with everywhere dense set ofbranch points (see, e.g., [2]).Let f : X → X be a continuous map of a dendrite X. There are theorems and examples ofcontinuous maps on dendrites showing that dynamics of such maps depends on the structure ofdendrites (see, e.g., [3] – [6]).In the report the correlation between the structure of dendrites and dynamics of continuous mapson them is investigated.References
[1] R. Kuratowski, Topology, V. 2, Academic Press, 1966.
[2] S. Nadler, Continuum Theory, Marcel Dekker, N.Y., 1992.
[3] J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology and its Applications, 144(2004),109-132.
[4] E.N. Makhrova, The structure of dendrites with the periodic point property, Russian Math. (Iz.VUZ), 55(2013), 33-37.
[5] E.N. Makhrova, Structure of dendrites admitting an existence of arc horseshoe, Russian Math.(Iz. VUZ), 59 (2015), 52–61.
[6] H. Abdelli, H. Marzougui, Recurrence and almost periodicity on dendrites, Chaos Solitons andFractals, 87 (2016), 17-18.
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VCDS-8 Talks 17
Recent results about sensitivity in general semiflows
Alica Miller
University of Louisville
We will talk about some results and questions from our recent paper about various types ofsensitivity in general semiflows (Applied General Topology 2018). We will also discuss resultsfrom recent papers on this topic by: (1) R. Thakur and R. Das; (2) X. Wu, X. Me, G. Chen, andT. Lu; and (3) X. Wu and X. Zhang.
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Renormalization towers and their forcing
Michał Misiurewicz
Indiana University-Purdue University Indianapolis
Over half a century ago, Sharkovsky proved his theorem about periods of periodic orbits of con-tinuous interval maps. Existence of some periods force existence of some other periods, and theordering obtained in such a way is linear. Later, people noticed that instead of looking at periods,one can take into account permutations. Unfortunately, this gives only a partial order, which isvery complicated and impossible to describe in simple terms. We propose the middle ground:looking at the block structures of permutations (they can be also understood in terms of renor-malizations). This is a finer classification of periodic orbits than just by periods, but still resultsin a linear ordering.This is joint work with Alexander Blokh.
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VCDS-8 Talks 18
Properties of Birkhoff spectra for the generic continuous functions on a shift space
Ryo Moore
Pontificia Universidad de Catolica de Chile
This is a joint work with Zoltan Buczolich and Balazs Maga. Let (Ω, σ) be the full-shift of twoalphabets, and f be a continuous, real-valued function on it. Let Lf be the set of all of the possiblelimiting values of the Birkhoff averages of f , i.e.
Lf :=
α ∈ R : ∃ω ∈ Ω such that limN→∞
1N
N−1∑
n=0
f(σnω) = α
.
For each α ∈ Lf , we define the level set
Ef (α) :=
ω ∈ Ω : limN→∞
1N
N−1∑
n=0
f(σnω) = α
,
and we define a function Sf : R→ R, which we refer to as the Birkhoff spectra, as follows:
Sf(α) :=
dimH(Ef (α)) α ∈ Lf ,0 α /∈ Lf ,
where dimH is the Hausdorff dimension.In this talk, we will discuss shapes and properties of the Birkhoff spectrum Sf for generic/typicalcontinuous functions f in the sense of Baire category. In particular, we will be interested in thebehavior of the spectrum near the boundary of Lf , such as continuity and values of one-sidedderivatives.
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Remarks on definitions of periodic points for nonautonomous dynamical system
Vojtech Pravec
Silesian University in Opava
Let (X, f1,∞) be a nonautonomous dynamical system. In this talk we summarize known definitionsof periodic points for general nonautonomous dynamical systems and propose a new definition ofasymptotic periodicity. This definition is not only very natural but also resistant to changes of abeginning of the sequence generating the nonautonomous system. We show the relations amongthese definitions and discuss their properties. We show that for uniformly convergent nonauto-nomous systems topological transitivity together with dense set of asymptotically periodic pointsimply sensitivity. We also show that even for uniformly convergent systems the nonautonomousanalog of Sharkovsky’s Theorem is not valid for most definitions of periodic points.
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VCDS-8 Talks 19
On mixing properties for a class of expanding Lorenz maps
Peter Raith
Universitat Wien
Consider a function f : [0, 1]→ [0, 2] which is continuous and strictly increasing. Assume that f isdifferentiable on (0, 1)\F where F is a finite set. Moreover assume that β := infx∈(0,1)\F f ′(x) > 1.This implies the existence of a unique c ∈ (0, 1) with f(c) = 1. Define Tfx := f(x)−⌊f(x)⌋, where⌊y⌋ is the largest integer smaller than or equal to y. One calls a map Tf of this form an expandingLorenz map. It is a piecewise monotonic map but it has a discontinuity at c.For β 3
√2 the map Tf is topologically transitive if f(0) 1
β+1. In the case β <
√2 it is
also topologically mixing unless f(x) = 3√2x + 2+
3√4−2 3√2
2for all x ∈ [0, 1]. Better results can be
obtained in the special case f(x) = βx+ α. Here one can completely describe the set of all (β, α)with 3
√2 ¬ β ¬ 2 and 0 ¬ α ¬ 2 − β such that Tf is topologically transitive and topologically
mixing.According to a definition of Glendinning the map Tf is called locally eventually onto if for everynonempty open U ⊆ [0, 1] there exist open intervals U1, U2 ⊆ U and n1, n2 ∈ N such that Tf
n1
maps U1 homeomorphically to (0, c) and Tfn2 maps U2 homeomorphically to (c, 1). Moreover, Tf
is called renormalizable if there are 0 ¬ u1 < c < u2 ¬ 1 and l, r ∈ N with l+ r 3 such that Tf lis continuous on (u1, c), Tf r is continuous on (c, u2), limx→c− Tf l x = u2 and limx→c+ Tf rx = u1.For expanding Lorenz maps relations between the notions “locally eventually onto” and “renor-malizable” are investigated.
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Mass transference principle for general shapes
Michal Rams
Mathematical Institute of the Polish Academy of Sciences
The mass transference principle is a theorem of Beresnevich and Velani describing the Hausdorffdimension of a class of fractals. More precisely, let (Bi) be a family of euclidean balls in [0, 1]d andassume Leb lim supBi = 1. Let Ei be a ball with the same center as Bi but with diameter |Ei|afor some a > 1. Then dimH lim supEi d/a.What I will present is a generalization of this theorem to the case when the sets Ei are just opensubsets of corresponding Bi, without conditions on shape or size. This is a joint work with HennaKoivusalo.
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VCDS-8 Talks 20
Uniformity of cocycles for subshifts with leading sequences
Daniel Sell
Friedrich-Schiller-Universitat Jena
Roughly speaking, a cocycle on a subshift Ω ⊆ AZ is a product of SL(2,R)-matrices, whose orderis determined by the letters in ω ∈ Ω. Such products appear for example in the study of Jacobioperators on subshifts. More specifically, it is known that the spectrum of a Jacobi operator is aCantor set of Lebesgue measure zero if the associated cocycle is uniform.This talk is concerned with a class of subshifts, in which “good convergence behaviour” of cocyclesalong arbitrary elements is guaranteed by “good behaviour” along finitely many of them (calledthe leading sequences). For these subshifts we show that every locally constant cocycle is uniform.We highlight the main ideas of the proof and discuss our most important example, simple Toeplitzsubshifts. The talk is based on joint work with Rostislav Grigorchuk, Daniel Lenz and TatianaNagnibeda.
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Vector-Valued Spectral Analysis of Complex Flows
Joanna Slawinska
Department of Physics, University of Wisconsin-Milwaukee
We introduce a recently developed framework for spatiotemporal pattern extraction called Vector-Valued Spectral Analysis (VSA). This approach is based on the eigendecomposition of a kernelintegral operator acting on vector-valued observables (spatially extended fields) of the dynamicalsystem generating the data, constructed by combining elements of the theory of operator-valuedkernels for multitask machine learning with delay-coordinate maps of dynamical systems. A keyaspect of this method is that it utilizes a kernel measure of similarity that takes into account bothtemporal and spatial degrees of freedom (whereas classical techniques such as EOF analysis arebased on aggregate measures of similarity between ’snapshots’). As a result, VSA has high skillin extracting physically meaningful patterns with intermittency in both space and time, whilefactoring out any symmetries present in the data. We demonstrate the efficacy of this methodwith applications to various cases of complex turbulent flows.
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VCDS-8 Talks 21
On the problem of characterizing minimal spaces/sets
Lubomır Snoha
Matej Bel University
Minimal sets are the fundamental objects of study in topological dynamics. The classification ofcompact minimal sets is a largely unsolved problem. Only for special classes something can besaid. Unsolved is also the problem as to which (compact) metric spaces admit minimal maps.We will survey some known related facts and we will present some recent results.
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Entropy and gluing orbit property
Peng Sun
Central University of Finance and Economics
Gluing orbit property is a much weaker form of specification property. It was introduced just a fewyears ago to study a broader class of non-uniformly hyperbolic systems. However, a system withgluing orbit property may have zero topological entropy, which differs from the case of specification.In this talk, we show that a system has gluing orbit property and zero topological entropy if andonly it is minimal and equicontinuous. Moreover, there is no minimal system that has both gluingorbit property and positive topological entropy. Some related results will also be discussed.
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Heat Conduction from Hamiltonian Dynamics
Domokos Szasz
Budapest University of Technology
Attempts to derive Fourier law of heat conduction from Hamiltonian dynamics have a historyduring which in 2008 Gaspard and Gilbert came up with a billiard model and suggested a twostep approach for it: 1. for the energies of the particles derive – in the rare interaction limit –a Markov jump process (dynamical part); 2. take the hydrodynamic limit of the obtained jumpprocess; this is expected to indeed lead to the Laplace equation (stochastic part). Since theirmodel was still unsuitable for mathematics, we introduced its tractable variant: the disk-pistonmodel. For it we can show that its rare interaction limit is, indeed, a Markov jump process (workin progress). The talk is based on joint works with P. Balint, Th. Gilbert, P. Nandori, IP. Tóth.
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VCDS-8 Talks 22
Permutation Doubling in One-Dimensional Dynamical Systems
Zoltan Szekely
University of Guam
Permutation doubling on a finite set, from an algebraic point of view, is a process when theunderlying set of the permutation is replaced by 2-element blocks, thus doubling the size of theunderlying set. The doubled permutation acts on the blocks the same way as the original permuta-tion acts on elements. Now, each 2-element block can be mapped onto another 2-element block twodifferent ways. This gives some flexibility in defining a given double of the original permutation.Thus, for any permutation there is a set of doubles or doubled permutations.We consider cyclic permutations on the underlying set 0, 1, 2, ..., n-1, where any i is mappedto i+1 modulo n. Shortly, we call these permutations naturally ordered cyclic permutations. Theusual induced real valued mapping f from I= [0, n-1] to I follows the rule f(i) = i+1 modulo n wheni is an integer in I and the values of f between integers are obtained by ”connecting-the-dots”,connecting the values given at integers by straight line segments. By iterating f over I, we countthe number of monotone line segments or laps of the iterated mapping. The corresponding lapnumber function l(t), as a function of the iteration number t, follows a simple recurrence relationof the Fibonacci fashion. Thus, depending on n, we obtain 2-bonacci (=Fibonacci), 3-bonacci, ...,(n-1)-bonacci relations. The last one is l(t) = l(t-1)+l(t-2)+...+l(t-n+1).In this talk we describe the recurrence relations obtained by doubling a naturally ordered cyclicpermutation. We’ll see that the (n-1)-bonacci relation is basically preserved by an added constant.The added constant depends on which individual double is taken from the set of doubles. Theresult allows generalization from doubles of naturally ordered cyclic permutations to doubles ofsome other permutations, as well.
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On Spectrum of Piecewise monotonic maps with Markov condition
Jan Tesarcık
Silesian Univerzity in Opava
We investigated Piecewise monotonic maps with Markov condition, namely Spectrum of Distribu-tional functions and properties of omega-limit sets. By Piecewise monotonic maps we mean mapf from interval I to itself, with a decomposition of I to finitely many intervals Jk such that f ison every Jk continuous and strictly monotone.
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VCDS-8 Talks 23
On topological classification of PCF self-similar dendrites
Andrei Tetenov
Novosibirsk state university/Gorno-Altaisk state university
By m-sprout we mean a finite tree Γ(B,W,E, ν, ϕ), whose set of vertices is divided into 2 partsV = B ∪ W , which is furnished with (injective) index function ν : 1, ..., m → B and edgecoloring function ϕ : E → 1, ..., m.
We define a composition operation Γ1 ∗ Γ2 of m-sprouts providing a semigroup structure on theset Sp(m) of all m-sprouts.
The powers Γn of a m-sprout Γ give rise to the sequence of topological spaces Xn, associated withΓn. If the sprout Γ is regular, the inverse limit X(Γ) = lim←−Xn is a post-critically finite dendrite,satisfying the condition X(Γ) =
⋃
w∈Wfw(X(Γ))
We prove the followingTheoremEach self-similar p.c.f. dendrite K is equivariantly homeomorphic to a dendrite X(Γ) defined bysome m-sprout Γ.
Thus, in postcritically finite case most of topological and metric properties of a self-similar dendriteare completely determined by the parameters of certain initial finite tree Γ, called m-sprout andwe explain in detail how these properties are determined by the m-sprout parameters.
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The Assouad spectrum and fine information on extremal scaling
Sascha Troscheit
University of Vienna
The Assouad spectrum was recently introduced by Fraser and Yu as an interpolation dimensionbetween the upper box-counting and Assouad dimension. As it turns out, it only interpolatesbetween the upper box-counting and quasi-Assouad dimension but gives fine information on scalingof sets. I will motivate its definition with several examples and give some applications. In the casewhere Assouad and quasi-Assouad dimensions differ, we will consider a natural extension of theAssouad spectrum definition and give threshold functions for which there is a “jump” in behaviour.
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VCDS-8 Talks 24
Complexity of dynamical systems
Benjamin Vejnar
Charles University
We study the complexity of conjugacy relation between dynamical systems (DS) using the methodsof invariant descriptive set theory. In this context, equivalence relations on Polish spaces arecompared by Borel reductions. We provide an overview of known results dealing with (minimal)Cantor DS, interval invertible DS, all DS. Additionally, we sketch the proof that interval DS canbe classified by countable structures.
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Bernoulli Disjointness
Benjamin Weiss
Hebrew University of Jerusalem
In his paper on disjointness (1967) Hillel Furstenberg proved that the Bernoulli shift 0, 1Z isdisjoint from every minimal Z-flow. We generalize this to all infinite discrete groups G. Namelythe G-shift on 0, 1G is disjoint from every minimal G-flow. Using this we show that the algebragenerated by the minimal functions on G is a proper subalgebra of l∞(G). These and several otherrelated results will be presented. This is joint work with Eli Glasner, Todor Tsankov and AndyZucker.
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VCDS-8 Talks 25
Entropy Black Hole
Wenxiang Sun
Peking University
I will talk about entropy black hole based on our series papers on topological entropy of flowswith singularity. There is a pair of equivalent flows so that one has zero entropy and another haspositive entropy, and furthermore, there is a pair of equivalent flows so that one has zero entropyand another has infinite entropy. There is a 4-D manifold M and a family of C∞ flows on M withexact one singularity so that all the flows share the same orbits. Flows are distinguished by speeds,and all flows but one with critical low speed have positive entropy. We deduce speed around thesingularity and thus deduce entropy, and at the end we get a zero entropy flow with the criticallow speed. We explain vividly that all entropies of the family of flows are “absorbed” by the flowwith the critical low speed, or are “absorbed ”by the singularity of this flow. We call the flowwith the critical low speed or just the singularity of this flow Entropy Black Hole comparing somecharacterizations with physical black hole.
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Geometry and computability at zero temperature
Christian Wolf
The City College of New York
Let f : X → X be a shift map. In this talk, we discuss questions concerning the geometryand computability of the space of continuous potentials φ : X → R with respect to certainthermodynamic invariants. In particular, we consider the residual entropy (i.e., the joint groundstate entropy) and the zero-temperature measure (i.e., the limit of equilibrium states when thetemperature goes to 0) of the system. We show that residual entropy h∞,φ is an upper semi-computable function of the potential, but it is not computable. Moreover, h∞,φ is computableat φ if and only if h∞,φ = 0. Next, we consider locally constant potentials for which the zero-temperature measure µ∞,φ is known to exist. We obtain a geometric characterization of the spaceof locally constant potentials in terms of their zero-temperature measure. In particular, we showthe existence of finitely many open convex cones of locally constant potentials of a fixed cylinderlength for which the zero-temperature measure can be computationally identified as an elementaryperiodic point measure. Finally, we show that our methods do not generalize to treat the case whenthe cylinder length is not given. The results presented in this talk combine joint work with YunYang (TAMS, to appear) and Michael Burr (arXiv:1809.00147).
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VCDS-8 Posters 26
Reduced approximation for opirator with time in parabolic problems case
Ali Berkane
High school of Management sciences
This work is devoted to singular perturbation of the parabolic equation with discontinuous coeffi-cients for the time operator. For P1-P0 finite element, by using a reduction of the approximationorder for the time differential operator, we propose a numerical method which does not have anyoscillations in the neighborhood of the coefficient discontinuity. Error estimates of order tow withrespect to space are provided, and we have compared this method with the modified second mem-ber method (T.T. Cuc Bui, 2008). Euler explicit and implicit time schemes are proposed, and byconsidering a toy problem, the order one and tow of convergence with respect to time and spaceis checked.
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Existence of solutions for delay dynamic equations on time scales
Faycal Bouchelaghem
High School of Management Sciences
In this work, we study the existence of positive soluttions for a dynamic equations on time scales.The main tool employed here is the Schauder fixed point theorem. The asymptotic properties ofsolutions are also treated. Three examples are also given to illustrate this work.
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Relation between Sensitive system and MDS using Furstenberg family
Mauricio Dıaz
Universidad del Bio Bio
This person also gives a talk, see the abstract in the talks section.
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VCDS-8 Posters 27
On logarithmic empirical measures and their limits
Habibeh Pourmand
Jagiellonian University
We study properties of logarithmic empirical measures in dynamical systems with specificationproperty. These are measure analogous to the classical empirical measures along orbit segmentsexcept that in the definition the logarithmic average replaces arithmetic average. These averageswere recently been used by Gomilko, Kwietniak and Lemańczyk to prove that Sarnak’s conjectureimplies Chowla’s conjecture along a subsequence. We prove that under a weak form of the specifi-cation property every invariant measures has a harmonic generic points. In fact, we prove a moregeneral result stating that for every nonempty, connected, and closed set V of invariant measuresthere is a dense set of orbits which have V as the limit points for logarithmic empirical measuresdefined along the orbit. This is a joint work with Dominik Kwietniak.
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Vector-Valued Spectral Analysis of Complex Flows
Joanna Slawinska
Department of Physics, University of Wisconsin-Milwaukee
This person also gives a talk, see the abstract in the talks section.
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Singular stationary measures for random piecewise affine interval homeomorphisms
Adam Śpiewak
University of Warsaw
We consider a family of random dynamical systems, each consisting of two piecewise affineincreasing homeomorphisms f−, f+ of the unit interval, each with exactly one point of non-differentiability, iterated randomly with probabilities (p1, p2). Since systems of this type wereintroduced in [1] by Alseda and Misiurewicz, we call them Alseda-Misiurewicz systems, or AM-systems. Under certain assumptions, such a system admits a unique stationary probability measureµ with no atoms at the endpoints. In this case, µ has to be either singular or absolutely continuouswith respect to the Lebesgue measure. We prove that µ is singular for a certain open set of para-meters, verifying a conjecture by Alseda and Misiurewicz [1] in this case. We also prove singularityand calculate Hausdorff dimensions of the measure µ and its support for systems satisfying someresonance conditions. This is joint work with Krzysztof Barański.
[1] L. Alseda, M. Misiurewicz, Random interval homeomorphisms, Publ. Mat., 58, (2014)
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