On orthonormal sets in inner product quasilinear spaces

Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math., 2 (1986), 23–52] generalized linear spaces by introducing the notion of quasilinear spaces in 1986. Then, special quasilinear spaces which are called ”solid floored quasilinear spaces” were defined and their some properties examined in [C¸ akan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, ˙Inon¨ u University, Malatya, 2016]. In fact, this classification was made so as to examine consistent and detailed some properties related ¨ to quasilinear spaces. In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. At the same time, the mentioned classification is crucial for define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems. Also, we try to explain some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.

Existence of generic cubic homoclinic tangencies for Hénon maps

In this paper, we show that the Hénon map $\varphi _{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko, Shilnikov and Turaev [On models with non-rough Poincaré homoclinic curves. Physica D 62(1–4) (1993), 1–14; Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6(1) (1996), 15–31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math.216 (1997), 70–118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69–128, translation in J. Math. Sci. 105 (2001), 1738–1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241–275]. Combining this fact with theorems in Kiriki and Soma [Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105–1140], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

Prevalence of rapid mixing—II: topological prevalence

We continue the study of mixing properties of generic hyperbolic flows started in an earlier paper (D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Erg. Th.& Dyn. Sys.18 (1998), 1097–1114). Our main result is that generic suspension flow over subshifts of finite type is exponentially mixing. This is a quantitative version of an earlier result of Parry and Pollicott (W. Parry and M. Pollicott. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst.216 (1997), 354–363).