Shrinking targets in parametrised families

Abstract

Abstract. We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdor dimension of the set of parameters for which the orbit of a xed point has a certain shrinking target property. This generalises several similar results for -transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.IF Introduction vet T X M 3 M e dynmil systemF sn nlogy to hiophntine pproximtionD rill nd elni II studied the set of points x P M suh tht the orit hits shrinking trget round nother point y innitely oftenF wore preiselyD they studied the set f x P M X d@T n @xA; yA < r n iFoFg; where r n 3 H s n 3 IF sn their seD T is n expnding rtionl mp on the iemnn sphereD nd M is the orresponding tuli setF hey lulted the rusdor dimension of the set in the se r n a e n F his nd similr sets hve lter een studied in severl dierent settingsD for instne for Etrnsformtions y fugeud nd ng T nd fugeud nd vio SF yne n lso study other relted setsD suh s f y P M X d@T n @xA; yA < r n iFoFg; for xed xD s ws done for instne in VD IT nd PIF sn this pperD we will onsider fmily of mps T a D where a is prmeterF por xed x nd yD we onsider the set of prmeters a for whih the orit of x hits shrinking trget round y innitely oftenD tht isD we onsider sets of the form f a X d@T n a @xA; yA < r n iFoFg: ets of this kind were previously studied y ersson nd hmeling PPD in the se where T a re EtrnsformtionsD T X H; I 3 H;IAY x U 3 x mod I; @ > IA; nd x a I nd y a HF he motivtion for studying the set E a f X T @IA a jT @IA Hj n iFoFg me from the question how well the dynmis of T n e pproximted with sushifts of nite typeF

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

“…For βtransformations such results were obtained by Bugeaud and Wang [BW14], and by Bugeaud and Liao [BL16]. Aspenberg and Persson [AP19] obtained some results for piecewise expanding maps that are not necessarily Markov maps. The Hausdorff dimension of sets of the form E(x, r n ) when T : x → 2x mod 1 was calculated by Fan, Schmeling and Troubetzkoy [FST13].…”

The Dolgopyat inequality in bounded variation for non-Markov maps

“…Persson for informing us about reference [2]. We are also grateful for the support the Erwin Schrödinger Institute in Vienna, where this paper was completed.…”

On the Intersection of Dynamical Covering Sets with Fractals

“…Bugeaud and Wang [5] studied the problem for the case when T is the β-transformation. Aspenberg and Persson [1] extended their results to piecewise expanding maps.…”

“…). Let (X, d) be a metric space with the finite doubling property and let 0 < b < 1 3 be a constant. Then there exists a collection {Q k,i : k ∈ Z, i ∈ N k ⊂ N} of Borel sets that have the following properties:…”

“…The distribution of the orbit of a point in X is an important topic in ergodic theory and has been studied by many authors. See, for example, [1,2,5,9,10,13]. The well-known Poincaré recurrence theorem shows that µ-a.e.…”