| Summary: | The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are <inline-formula> <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> </inline-formula>-shifts, namely transformations of the form <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo lspace="0pt">:</mo> <mi>x</mi> <mo>↦</mo> <mi>β</mi> <mi>x</mi> <mo>+</mo> <mi>α</mi> <mspace width="0.277778em"></mspace> <mo form="prefix">mod</mo> <mspace width="0.277778em"></mspace> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> acting on <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">[</mo> <mo>−</mo> <mi>α</mi> <mo>/</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>/</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo>Δ</mo> </mrow> </semantics> </math> </inline-formula> is fixed and where <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>≔</mo> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> <mo lspace="0pt">:</mo> <mi>β</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="0.277778em"></mspace> <mi>and</mi> <mspace width="0.277778em"></mspace> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> <mo>−</mo> <mi>β</mi> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> such that <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </semantics> </math> </inline-formula> has the subshift of finite type property is dense in the parameter space <inline-formula> <math display="inline"> <semantics> <mo>Δ</mo> </semantics> </math> </inline-formula>. Here, they proposed the following question. Given a fixed <inline-formula> <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> which is the <i>n</i>-th root of a Perron number, does there exists a dense set of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> in the fiber <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <mi>β</mi> <mo>}</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>, so that <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </semantics> </math> </inline-formula> has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> to the case when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>. That is, we examine the structure of the set of eventually periodic points of <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </semantics> </math> </inline-formula> when <inline-formula> <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> </inline-formula> is a Pisot number and when <inline-formula> <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> </inline-formula> is the <i>n</i>-th root of a Pisot number.
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