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Minor cycles for interval maps

Michał Misiurewicz (1994)

Fundamenta Mathematicae

For continuous maps of an interval into itself we consider cycles (periodic orbits) that are non-reducible in the sense that there is no non-trivial partition into blocks of consecutive points permuted by the map. Among them we identify the miror ones. They are those whose existence does not imply existence of other non-reducible cycles of the same period. Moreover, we find minor patterns of a given period with minimal entropy.

Misiurewicz maps unfold generically (even if they are critically non-finite)

Sebastian van Strien (2000)

Fundamenta Mathematicae

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if f λ 0 is critically finite with non-degenerate critical point c 1 ( λ 0 ) , . . . , c n ( λ 0 ) such that f λ 0 k i ( c i ( λ 0 ) ) = p i ( λ 0 ) are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31   λ ( f λ k 1 ( c 1 ( λ ) ) - p 1 ( λ ) , . . . , f λ k d - 2 ( c d - 2 ( λ ) ) - p d - 2 ( λ ) ) is a local diffeomorphism...

Mixed formulation for elastic problems - existence, approximation, and applications to Poisson structures

Julian Ławrynowicz, Alain Mignot, Loucas Papaloucas, Claude Surry (1996)

Banach Center Publications

A mixed formulation is given for elastic problems. Existence and uniqueness of the discretized problem are given for conformal continuous interpolations for the stress tensor components and for the components of the displacement vector. A counterpart of the problem is discussed in the case of an even-dimensional Euclidean space with an associated Hamiltonian vector field and the Poisson structure. For conformal interpolations of the same order the question remains open.

Mixing on rank-one transformations

Darren Creutz, Cesar E. Silva (2010)

Studia Mathematica

We prove that mixing on rank-one transformations is equivalent to "the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums". In particular, all polynomial staircase transformations are mixing.

Mixing properties of nearly maximal entropy measures for d shifts of finite type

E. Robinson, Ayşe Şahin (2000)

Colloquium Mathematicae

We prove that for a certain class of d shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any A , . . . , A k of positive...

Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada (2004)

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by d ( x ) : = i = 1 n 1 E ( 2 i - 1 x ) ( m o d 2 ) , where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if N - 1 n = 1 N d ( x ) converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that N - 1 n = 1 N d ( x ) converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Model of AIDS-related tumour with time delay

Marek Bodnar, Urszula Foryś, Zuzanna Szymańska (2009)

Applicationes Mathematicae

We present and compare two simple models of immune system and cancer cell interactions. The first model reflects simple cancer disease progression and serves as our "control" case. The second describes the progression of a cancer disease in the case of a patient infected with the HIV-1 virus.

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