scholarly journals Volume hyperbolicity and wildness

2016 ◽  
Vol 38 (3) ◽  
pp. 886-920
Author(s):  
CHRISTIAN BONATTI ◽  
KATSUTOSHI SHINOHARA
Keyword(s):  
Main Tool ◽  
Periodic Points ◽  
Necessary Condition ◽  
Partial Hyperbolicity ◽  
Chain Recurrence ◽  
Markov Partitions ◽  
Open Set ◽  
Partially Hyperbolic ◽  
Uniform Expansion

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$-manifold $M$, the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

scholarly journals A link between topological entropy and Lyapunov exponents

2017 ◽  
Vol 39 (3) ◽  
pp. 620-637
Author(s):  
THIAGO CATALAN
Keyword(s):  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Lower Semicontinuous ◽  
Periodic Points ◽  
Partial Hyperbolicity ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism ◽  
Partially Hyperbolic ◽  
Generic Set ◽  
Hyperbolic Sets

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

Partially hyperbolic and transitive dynamics generated by heteroclinic cycles

2001 ◽  
Vol 21 (1) ◽  
pp. 25-76 ◽  
Author(s):  
LORENZO J. DÍAZ ◽  
JORGE ROCHA
Keyword(s):  
Periodic Points ◽  
Heteroclinic Cycles ◽  
Hyperbolic Dynamics ◽  
Open Set ◽  
Transitive Set ◽  
Locally Maximal ◽  
Partially Hyperbolic ◽  
Homoclinic Tangencies

We study \mathcal{C}^k-diffeomorphisms, k\ge 1, f: M\to M, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). We prove that if f can not be \mathcal{C}^k-approximated by diffeomorphisms with homoclinic tangencies, then f is in the closure of an open set \mathcal{U}\subset \operatorname{Diff}^k(M) consisting of diffeomorphisms g with a non-hyperbolic transitive set \Lambda_g which is locally maximal and strongly partially hyperbolic (the partially hyperbolic splitting at \Lambda_g has three non-trivial directions). As a consequence, in the case of 3-manifolds, we give new examples of open sets of \mathcal{C}^1-diffeomorphisms for which residually infinitely many sinks or sources coexist (\mathcal{C}^1-Newhouse's phenomenon). We also prove that the occurrence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

scholarly journals Partial hyperbolicity and classification: a survey

2016 ◽  
Vol 38 (2) ◽  
pp. 401-443 ◽  
Author(s):  
ANDY HAMMERLINDL ◽  
RAFAEL POTRIE
Keyword(s):  
Fundamental Group ◽  
Three Dimensional ◽  
Group Classification ◽  
Partial Hyperbolicity ◽  
Hyperbolic Diffeomorphisms ◽  
Higher Dimensional ◽  
Partially Hyperbolic ◽  
Anosov Flows ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Solvable Fundamental Group

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

scholarly journals Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Jerzy Jezierski
Keyword(s):  
Lie Groups ◽  
Explicit Form ◽  
Homotopy Class ◽  
Unitary Group ◽  
Compact Manifold ◽  
Periodic Points ◽  
Necessary Condition ◽  
Minimal Realization ◽  
Compact Lie Groups

AbstractLet $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

scholarly journals Simulating the Impact of Income Distribution on Poverty Reduction

2015 ◽  
Vol 54 (4I-II) ◽  
pp. 931-944
Author(s):  
Syed Kalim Hyder ◽  
Qazi Masood Ahmed ◽  
Haroon Jamal
Keyword(s):  
Economic Growth ◽  
Income Distribution ◽  
Poverty Alleviation ◽  
Poverty Reduction ◽  
Main Tool ◽  
High Growth ◽  
Quality Data ◽  
Necessary Condition ◽  
The Impact ◽  
Trickle Down

The traditional notion that has influenced the development thinking for almost half a century is that economic growth is fundamental to the development process, and that the objective of poverty reduction can only be achieved by allowing the benefits of growth to ultimately trickle down to the poor. The „primacy of growth‟ paradigm is based on the premise that high growth, through high investment, would lead to higher employment and higher wages, and thereby reducing poverty. The „trickle-down‟ paradigm assumes that the benefits of economic growth would, in the first round, accrue to the upper income groups, and the ensuing consumption expenditures of these households would, in subsequent rounds, accrue incomes to relatively lower income households. Importance of equity consideration in poverty alleviation efforts has been brought out of the cold and now has re-entered the mainstream development policy agenda in many developing countries. This is the consequence of a deep-rooted disillusionment with the development paradigm which placed exclusive emphasis on the pursuit of growth. During 1990s, the proliferation of quality data on income distribution from a number of countries has allowed rigorous empirical testing of standing debates on the relative importance of growth and redistribution in poverty reduction. While the debate is still inconclusive, the majority of development economists emphasised, based on empirical cross-country data, that an unequal income distribution is a serious impediment to effective poverty alleviation [Ravallion (1997, 2001)]. Many researchers suggested that growth is, in practice the main tool for fighting poverty. However, they also reiterated that the imperative of growth for combating poverty should not be misinterpreted to mean that “growth is all that matters”. Growth is a necessary condition for poverty alleviation, no doubt, but inequality also matters and should also be on the development agenda

Controlled Lotka-Volterra systems in the modeling of biomedical processes

2021 ◽  
Author(s):  
Evgenii Khailov ◽  
Nikolai Grigorenko ◽  
Ellina Grigorieva ◽  
Anna Klimenkova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Numerical Study ◽  
Treatment Strategies ◽  
Main Tool ◽  
Necessary Condition ◽  
Control Problems ◽  
Optimal Controls ◽  
Controlled Systems ◽  
Volterra Systems

This book is devoted to a consistent presentation of the recent results obtained by the authors related to controlled systems created based on the Lotka-Volterra competition model, as well as to theoretical and numerical study of the corresponding optimal control problems. These controlled systems describe various modern methods of treating blood cancers, and the optimal control problems stated for such systems, reflect the search for the optimal treatment strategies. The main tool of the theoretical analysis used in this book is the Pontryagin maximum principle - a necessary condition for optimality in optimal control problems. Possible types of the optimal blood cancer treatment - the optimal controls - are obtained as a result of analytical investigations and are confirmed by corresponding numerical calculations. This book can be used as a supplement text in courses of mathematical modeling for upper undergraduate and graduate students. It is our believe that this text will be of interest to all professors teaching such or similar courses as well as for everyone interested in modern optimal control theory and its biomedical applications.

Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

2011 ◽  
Vol 32 (1) ◽  
pp. 63-79 ◽  
Author(s):  
J. BUZZI ◽  
T. FISHER ◽  
M. SAMBARINO ◽  
C. VÁSQUEZ
Keyword(s):  
Hopf Bifurcation ◽  
Maximal Entropy ◽  
Open Set ◽  
Entropy Measures ◽  
Unique Measure ◽  
Anosov Systems ◽  
Anosov System ◽  
Partially Hyperbolic ◽  
Structurally Stable ◽  
Measure Of Maximal Entropy

AbstractWe show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

ELLIPTIC NON-BIRKHOFF PERIODIC ORBITS IN THE TWIST MAPS

1993 ◽  
Vol 03 (01) ◽  
pp. 165-185 ◽  
Author(s):  
ARTURO OLVERA ◽  
CARLES SIMÓ
Keyword(s):  
Periodic Orbits ◽  
Periodic Points ◽  
Irrational Rotation ◽  
Point Of View ◽  
Twist Map ◽  
Qualitative And Quantitative ◽  
Open Set ◽  
Twist Maps ◽  
Rotational Curve ◽  
Irrational Rotation Number

We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches. In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

scholarly journals Homoclinic points, atoral polynomials, and periodic points of algebraic -actions

2012 ◽  
Vol 33 (4) ◽  
pp. 1060-1081 ◽  
Author(s):  
DOUGLAS LIND ◽  
KLAUS SCHMIDT ◽  
EVGENY VERBITSKIY
Keyword(s):  
Growth Rate ◽  
Main Tool ◽  
Periodic Points ◽  
Laurent Polynomials ◽  
Homoclinic Points ◽  
Complex Variety ◽  
Finite Set ◽  
Algebraic Actions ◽  
The Ideal

AbstractCyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

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