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Manual A Sequence of Problems on Semigroups (Problem Books in Mathematics)

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Ideally, puzzles can teach a productive new way of framing or representing a given situation. Although the border between the two is not always clearly defined, problems tend to require a more systematic application of formal tools, and to stress more technical aspects. Thus, a major aim of the present collection is to bridge the gap between introductory texts and rigorous state-of-the-art books. Anyone with a basic knowledge of probability, calculus and statistics will benefit from this book; however, many of the problems collected require little more than elementary probability and straight logical reasoning.

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JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. In that work, Desch et al introduced some computable conditions for hypercyclicity and Devaney chaos.

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One of them is stated in terms of the abundance of eigenvalues of the infinitesimal generator of the semigroup, see Theorem 2. This criterion allowed to extend several known examples whose solutions present a wild behaviour. In this paper we review some results concerning the linear dynamics of C 0 -semigroups of operators. In Section 1 we present several notions that have been considered in the study of linear dynamics of C 0 -semigroups. For the clarity of the exposition, the formulations of these notions for single linear operators are omitted. The existence of computable criteria for hypercyclicity and other dynamical properties is considered in Section 2.

One of the most well-known criteria in the area is the Desch, Schappacher, and Webb criterion in any of their formulations. We revisit different versions of the criterion, and some examples for which the criteria were applied. In the same way as the weighted shifts are considered as a model for understanding the dynamics of linear operators on sequence spaces [ 26 - 28 ], the translation semigroup works in the same manner for the dynamics of semigroups [ 20 , Sec.

This has been fostered part because of the use of the comparison lemma for extending the results to other semigroups. The characterization of the dynamics of the translation semigroup is considered in Section 3.


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A special case is provided by semigroups generated by partial differential equations in spaces of analytic functions with some growth control, which is analyzed in Section 4. The revision of solution semigroups of partial differential equations defined on a special Banach sequence space, the Herzog space, is provided in Section 5.

Finally, Section 6 deals with some open problems for future study on the dynamics of semigroups. In this section, we gather the most significant notions that have been considered in the study of linear dynamics of semigroups in the last years. First of all, we will recall the basic notions on linear dynamics. The study of hypercyclicity started with the work of Kitai [ 1 , 29 ].

Beauzamy coined the notion of hypercyclicity for linear operators, see [ 30 , 31 ]. Further information on the origins of this notion can be found in [ 5 ]. This concept is stronger than the ones of supercyclicity or cyclicity, in which one considers the multiples of the elements of the orbit or their linear combinations, respectively, instead of the orbit itself. In this work we will restrict ourselves to hypercyclicity.

It consists on the existence of elements with dense orbit. The generalization to semigroups is natural, replacing the discrete orbit of an element by a continuous one. A surprising result is that the orbit of a vector under a semigroup is dense as soon as it is somewhere dense see [ 32 ], based on a result of Bourdon and Feldman [ 33 ]. This notion is equivalent to transitivity. Hypercyclicity is a quite unstable property, since by small perturbations of the infinitesimal generator we can destroy it [ 35 ].

A detailed study of the equivalences between the dynamics of semigroups and the dynamics of their discretizations can be found in [ 36 ]. The particular case of autonomous discretizations, that is the existence of hypercyclic operators in the semigroup, requires a special attention. Costakis and Sambarino proved that such a set can be shared by all the non-trivial operators of the translation semigroup on the complex plane [ 37 ].

This also holds for supercyclic semigroups and their non-trivial operators [ 39 ]. However, these results cannot be extended to the frame of semigroups whose index set is a sector, or the whole, complex plane. The translation semigroup considered on a suitable L p -space may result to be hypercylic, or even mixing, but with no single operator satisfying such property [ 36 , Ex.

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The situation does not improve if we restrict ourselves to holomorphic semigroups [ 40 ]. After the aforementioned works of Birkhoff and Rolewicz, the study of the wild dynamics of semigroups was taken on by Lasota, who considered the existence of turbulent orbits in [ 21 ]. In fluid dynamics, turbulence is associated with low momentum diffusion, high momentum convection, and rapid variation of pressure and flow velocity in space and time. A solution is strictly turbulent if its closure is a compact non-empty set and it does not contain periodic points. Roughly speaking, they are complicated and irregular.

This was quite surprising since turbulence seemed to be tied to very strongly nonlinear partial differential equations of higher order. Following this line, in [ 41 ], Lasota also showed an example of a solution semigroup associated to the following Abstract Cauchy Problem that describes the growth of a population of cells which constantly differentiate change their properties in time.

The usual setting for posing these problems are the spaces of continuous or integrable functions on the interval. The existence of semigroups is not limited to partial differential operators in function spaces. It is known that every separable infinite-dimensional Banach space admits a hypercyclic semigroup [ 45 ]. The proof relies on a construction based on a result on biorthogonal sequences on Banach spaces by Ovsepian and Pelczynski [ 46 ] and on an analogous result for single operators [ 47 ]. An alternative shorter proof was given in [ 48 ].

6. Araoz's Semigroup Problem | Integer Programming | Society for Industrial and Applied Mathematics

Godefroy and Shapiro introduced the notion of chaos in the sense of Devaney [ 51 ] for linear operators [ 3 ]. In our setting, the last condition can be directly obtained from hypercyclicity. In the particular setting of linear operators the SDIC can be deduced from the mere hypercyclity [ 52 ]. In [ 53 ], following an ergodic-theoretical approach, Rudnicki showed the existence of invariant measures having strong analytic and mixing properties of the solution semigroup of equation 5.

He also showed the existence of Devaney chaos, see also [ 54 - 56 ].

An updated revision of these results can be found in [ 57 , 58 ]. The reader can find more information about the study of the dynamics of semigroups induced by semiflows in [ 64 ]. The notion of transitivity can be strengthened in some ways.

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The weak mixing property was considered in the setting of linear dynamics by Herrero [ 65 ]. He also posed there the question whether every hypercyclic operator was weakly mixing. This question was shown to be negative by De la Rosa and Read [ 66 ], see also [ 67 ]. However, the weakly mixing property was seen to be equivalent to the Hypercyclicity Criterion [ 68 ]. This link can be shown in the following result:. The topologically mixing property was first analyzed for single operators in [ 69 ], and later studied for semigroups in [ 70 ]. However, when all the discretizations of a semigroup satisfy either transitive, weakly mixing, or the mixing property, the other properties also hold.

With the goal of studying linear transformations from the point of view of ergodic theory, Bayart and Grivaux introduced the notion of frequent hypercyclicity in order to quantify the frequency with which an orbit meets open sets [ 73 ]. This notion was extended to semigroups in [ 74 ]. The set of frequent hypercyclic vectors for an operator is meager [ 75 ]. However, if one considers the use of the upper density in the definition of frequent hypercyclicity, the set of upper frequent hypercyclic vectors is residual [ 75 ].

In the celebrated paper of Li and Yorke [ 76 ], they introduced the concept of scrambled set. In this flavor, this notion was studied in linear dynamics in [ 77 ] and [ 78 ]. Irregular vectors were introduced by Beauzamy [ 79 ], and their existence turned out to be equivalent to Li-Yorke chaos [ 77 ].

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It was incorporated to the setting of linear dynamics in [ 81 , 82 ], and thoroughly studied in [ 83 ]. The corresponding version for C 0 -semigroups was given in [ 84 ].

In this notion we require a scrambled set S such that the orbits of any couple of distinct points are closed enough or at least at some distance, measured in terms of upper densities. Inspired by the concept of irregular vectors, a new notion of distributionally irregular vector was given in [ 77 ] for single operators, later generalized for C 0 -semigroups:.

The specification property was introduced by Bowen in [ 87 ] in order to indicate that there exist periodic points that can approximate prescribed parts of the orbits of certain elements. It is a very strong property, and it was stated for compact metric spaces. In the context of linear dynamics, the corresponding concept was given in [ 88 ], and thoroughly studied in [ 89 ] for single operators.

A recent adaptation to C 0 -semigroups was provided:.